Jay Gopalakrishnan University of Florida Montreal ...web.pdx.edu/~gjay/talks/montrealI.pdf · Jay...

Jay Gopalakrishnan

Multigrid methodsJay Gopalakrishnan

University of Florida

Montreal Scientific Computing Days

Centre de Recherches Mathematiques

February 2005

Minicourse Part I

Department of Mathematics [Slide 1 of 36]

Jay Gopalakrishnan

Why multigrid

Multigrid techniques give algorithms that solve sparse linearsystems

Ax = b

of N unknowns with O(N) work and storage for large classesof problems It applies typically to systems arising fromdiscretization of partial differential equations

Iterative method Convergence rate estimates

Gauszlig-Seidel δ lt 1minus Ch2

SOR δ lt 1minus Ch

ADI δ lt (1minus Ch)2

k-parameter ADI δ lt (1minus Ch1k)2

Multigrid δ lt 1 independent of h


Jay Gopalakrishnan

Why multigrid



SOR with best parameter δ lt 1minus Ch




Application minus∆u = f in Ω equiv (minus1 1)2 u = 0 on partΩ

Discretization method Linear finite elements on uniform grid of mesh-size h

Meaning of δ For many iterative methods one can prove that the iterates x(i)

satisfy x(i+1) minus x le δx(i) minus x for some 0 lt δ lt 1 in some norm middot


Jay Gopalakrishnan

Why multigrid


Gauszlig-Seidel [classical] δ lt 1minus Ch2

SOR [Young 1950] δ lt 1minus Ch

ADI [Peaceman amp Rachford 1955] δ lt (1minus Ch)2

k-parameter ADI [1960rsquos] δ lt (1minus Ch1k)2

Multigrid [see below] δ lt 1 independent of h

Source

Wesselingrsquos book 1992

60rsquos [Fedorenko 1964] [Bachvalov 1966]70rsquos [Brandt 1973] [Nicolaides 1975] [Brandt 1977]80rsquos [Bank amp Dupont1981] [Braess amp Hackbusch1983] [Bramble amp Pasciak1987]


Jay Gopalakrishnan

Structure of multigrid algorithmsMultigrid algorithms are based on a sequence of meshesobtained by successive refinement

Whenever it is possible to solve on the coarsest mesh fastmultigrid algorithms allow fast solution on the finest mesh

A 2D example

k = 1 k = 2

k = J

Highlyrefined

Mesh 1 Mesh 2 Mesh J

(Coarsest mesh) (Finest mesh)

Multigrid algorithms have a recursive structure Eachmultigrid iteration typically consists of the following steps

1 Smooth errors at current grid

2 Transfer residual to next coarser grid

3 Correct iterate using the coarser residual (recursively)


Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined

Need to solveAJu = b

Suppose discretization of some continuum problem at meshlevel k gives a linear operator Ak

To compute u = Aminus1J b iteratively we use multigrid iterations

u(i+1) = MgJ(u(i) b) i = 0 1 2

starting with some initial guess u(0) where the routineMgJ(middot middot) recursively invokes MgJminus1(middot middot) MgJminus2(middot middot)

We set Mg1(v b) equiv Aminus11 b

Idea

1 Reduce fine grid components of error

2 Reduce coarse grid components of error

Difficulty Donrsquot know exact solution so donrsquot know the error


Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined


Suppose discretization of some continuum problem at meshlevel k gives a linear operator AkIdea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined



1 Reduce fine grid components of errorBy smoothing error (without knowing the error)

2 Reduce coarse grid components of errorWe donrsquot have error e but we have residual r = bminusAJu(i) = AJe



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




2 Reduce coarse grid components of errorNeed an approximation for the coarse components of Aminus1

Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




2 Reduce coarse grid components of errorApply the routine Mg

Jminus1 to r projected to the next coarser grid



Jay Gopalakrishnan

A typical pseudocode

k = 1 k = 2

k = J

Highlyrefined


Suppose discretization of some continuum problem at meshlevel k gives a linear operator AkALGORITHM u(i+1) larrminus MgJ(u(i) b)

1 Smooth errors v = SmoothJ(u(i) b)

2 Transfer residual to coarser gridr = RestrictJminus1(bminus AJv)

3 Correct by recursion w = MgJminus1(0 r)

u(i+1) = v + ProlongJ(w)Department of Mathematics [Slide 5 of 36]

Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined



1 Smoothing v = SmoothJ(u(i) b)

2 Correction

u(i+1) = v + ProlongJ(MgJminus1(0 RestrictJminus1(bminus AJv)))


Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction

u(i+1) = v + LJMgJminus1(0 LtJ(bminus AJv))


Jay Gopalakrishnan

Eg 1 Laplace equation

FEM

Find uh isin Vh satisfying

(nablauhnablavh) = (f vh) forallvh isin Vh

Vh sub H10(Ω) is any standard fe space

Weakform

Find u isin H10 (Ω) satisfying

(nablaunablav) = (f v) forallv isin H10(Ω)

BVPminus∆u = f on Ω

u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator

Rewrite discrete problem as the operator eq

Ahuh = fh

where Ah Vh 7rarr Vh is defined by

(Ahwh vh) = (nablawhnablavh) forallwh vh isin Vh

Need multigrid to solve for uh equiv Aminus1h fh efficiently


Jay Gopalakrishnan

Eg 1 Multigrid setting

Assume that Vh is a fe space on a highly refined mesh

Ω

middot middot middot

V1 V2 VJ equiv Vh

Multilevel spacesVk = vh isin H

10(Ω) vh|K isin P1(K) for all elements K in

the kth level mesh

Multilevel operators At each level we also have operatorsgenerated by (nablamiddotnablamiddot) namely Ak Vk 7rarr Vk defined by

(Akv w) = (nablavnablaw) forallv w isin VkDepartment of Mathematics [Slide 7 of 36]

Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh

ALGORITHM u(i+1) larrminus MgJ(u(i) b)

1 Smoothing v = SmoothJ (u(i) b)

2 Correction

u(i+1) = v + LJ MgJminus1(0 LtJ (bminus AJ v))


Jay Gopalakrishnan

Eg 1 Prolongation

The multilevel spaces in this example are nested

V1 sub V2 sub middot middot middot sub VJ

Hence we choose Lk to be the imbedding operator

Vkminus1 rarr Vk

Computationally this means we simply implement a change ofbasis matrix

Ω

v1 isin V1 L2v1 isin V2


Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan

Elliptic eigenfunctionsThe smoothing component of multigrid relies on the fact thatthe eigenfunctions of elliptic operators corresponding to highereigenvalues are increasingly oscillatory

minus∆φ` = λ`φ` φ`L2(Ω) = 1

Eg here are the 1st 50th and 700th eigenfunctions of adiscrete Laplacian on an L-shaped domain

minus1 minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother

Let λ(k)max be the largest eigenvalue of (spd) Ak Then

ALGORITHM x(i+1) larrminus Smoothk(x(i) b)

x(i+1) = x(i) + (1λ(k)max)(bminus Akx

(i))

is the Richardson smootherWhy does this iteration smooth errors

If the eigenvalues of Ak are 0 lt λ1 le λ2 le middot middot middotλn equiv λ(k)max

and the eigenfunction corresponding to λ` is ψ` then

e = c1ψ1 + c2ψ2 + middot middot middot+ cnψn

=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max

)ψ1 + middot middot middot+ cn()ψn

=rArr The components in ψn-direction are annihilated + ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))


First observe the propagation of errors e(i)

x(i+1) = x(i) + (1λ(k)max)(Akxminus Akx

(i))

x = x + (1λ(k)max)(Akxminus Akx)

=rArr e(i+1) = e(i) minus (1λ(k)max)Ake

(i)

Hence an equivalent question is

why is I minus (1λ(k)max)Ak a smoothing operator




=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max

)ψ1+middot middot middot+cn(1minusλn

λ(k)max

)ψn

=rArr The components in ψn-direction are annihilated

+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max

)ψ1+middot middot middot+cn(1minusλ

(k)max

λ(k)max

)ψn



Jay Gopalakrishnan

Eg 1 The algorithmThus all components of the algorithm are now well defined



2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction

u(i+1) = v + LJ MgJminus1(0 LtJ (bminus AJv))

This algorithm is a ldquobackslash multigrid cyclerdquo (or ldquondashcyclerdquo)


Jay Gopalakrishnan

Eg 1 A V-cycle algorithm

ALGORITHM u(i+1) larrminus VcycleJ(u(i) b)

1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction

w = v + LJ VcycleJminus1(0 LtJ (bminus AJv))

3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan

Multigrid cyclesSchedule of multilevel grids in standard multigrid algorithms

-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1

All these cycles satisfy the optimal O(N) work estimateDepartment of Mathematics [Slide 13 of 36]

Jay Gopalakrishnan

Braess-Hackbusch theoremConsider the error reduction operator Ek given by

uminus u(i+1) = uminus Vcyclek(u(i) b) equiv Ek (uminus u(i))

Let a(u v) equiv (nablaunablav) and |||v|||a equiv a(v v)12

[Braess amp Hackbusch1983]

THEOREM If Ω is a convex polygon then the V-cycle algorithmfor Eg 1 converges at a rate δ lt 1 independent of mesh sizes

|||Ek|||a le δ


Jay Gopalakrishnan

ProofThe proof is difficult to motivate since it demands considerabletechnical ingenuity Here is an outline of its ldquomodernrdquo version

Step 1 With Kk = I minus (1λ(k)max)Ak we first prove

|||(I minus Pkminus1)Kkv|||2a le C

[

|||v|||2a minus |||Kkv|||2a

]

forallv isin Vk

where Pkminus1 is the orthogonal projection into Vkminus1 with respectto the a(middot middot)ndashinnerproduct

Vk = Pkminus1Vk︸︷︷︸

oplus (I minus Pkminus1)Vk︸︷︷︸

Coarse grid components Fine grid components


[


]

forallv isin Vk


Jay Gopalakrishnan



|||(I minus Pkminus1)Kkv|||2a︸︷︷︸le C

[


]

︸︷︷︸

forallv isin Vk

Norm of fine grid componentsafter smoothing

le CA measure of damp-ing by the smoother

Thus if a v isin Vk is left undamped by the smoother ie if

|||v|||a asymp |||Kkv|||a

then v must be a coarse grid function (roughly)


[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk

Step 2 The error reduction operator Ek of the algorithmsatisfies the recursion

a(Eku u)= |||(I minus Pkminus1)Kku|||2a+a(Ekminus1Pkminus1Kku Pkminus1Kku)

Using Step 1 and estimating we eventually prove the theorem


Jay Gopalakrishnan

Proof

Step 1 elaborated |||(I minus Pkminus1)w|||2a =

= (w minus Pkminus1wAkw)

= w minus Pkminus1wL2(Ω)

radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof



le w minus Pkminus1wL2(Ω)

radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]

Thus |||(I minus Pkminus1)w|||2a le

C hminus1k w minus Pkminus1wL2(Ω)

radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)

le C|||w minus Pkminus1w|||aby the Aubin-Nitscheduality argument forfinite elements


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]

Thus |||(I minus Pkminus1)w|||a le Cradic

a(ww)minus a(Kkww)

Using also the convergence properties of the smoothingiteration we finally have


[


]

forallv isin Vk

Step 2 elaborated Later Department of Mathematics [Slide 16 of 36]

Jay Gopalakrishnan

Regularity amp ApproximationA critical inequality in the previous proof is

w minus Pkminus1wL2(Ω) le Chk|||w minus Pkminus1w|||a

This is proved using the Aubin-Nitsche duality argument whichrequires that the exact solution φ of

minus∆φ = f on Ω φ = 0 on partΩ

has an approximation φk isin Vk satisfying

|||φminus φk|||a le ChkfL2(Ω)

This is known to hold when Ω is a convex polygon

|φ|H2(Ω) le CfL2(Ω) |||φminus φk|||a le Chk|φ|H2(Ω)

( REGULARITY + APPROXIMATION)Department of Mathematics [Slide 17 of 36]

Jay Gopalakrishnan

Practical smoothers

The Richardson smoother requires λ(k)max at every level k

These numbers are not easy to obtain in practice even forsimple examples

Fortunately many other classical iterative methods possessthe smoothing property

x(i+1) larrminus Jacobi(x(i) b)

x(i+1) larrminus Gauszlig-Seidel(x(i) b)


Jay Gopalakrishnan

The smoothing effectA classical linear iteration for a matrix A of the form

x(i+1) = x(i) + R(bminus Ax(i))

with some matrix R is a smoothing iteration if

(Iminus RA)e is smoother than e for any e


x = x + R(bminus Ax)

e(i+1) = e(i) minus RAe(i)

(Hence smoothing iterations smooth errors)


Jay Gopalakrishnan





If D is the diagonal and L is the lower triangular part of A then

Jacobi iteration R = Dminus1

Gauszlig-Seidel iteration R = (L + D)minus1

The smoothing effect of these iterations is easy to demonstratecomputationally (Click to see code)


Jay Gopalakrishnan

The smoothing effect

The smoothing effect on errors of Gauszlig-Seidel iteration

minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy

A random vector After 7 Gauszlig-Seidel iterations


Jay Gopalakrishnan

The smoothing conditionThe mathematical statement of the smoothing effect of anyiteration of the form

x(i+1) = x(i) +Rk(bminus Akx(i))

that is useful for multigrid analysis is as before


[


]

forallv isin Vk

but now with Kk = I minusRkAk

As in the case of the Richardson iteration proof of thistypically requires not only an analysis of smoother Rk but alsoelliptic regularity and approximation estimates


Jay Gopalakrishnan

Abstract V-cycleAssume general nested spaces and inherited bilinear forms

V1 sub V2 sub middot middot middot sub VJ sub middot middot middot ( V a(middot middot) )

A1 A2 middot middot middot AJ middot middot middot A

R1 R2 middot middot middot RJ

All spaces have a base innerproduct 〈middot middot〉 and norm middot Operator Ak satisfies 〈Akv w〉 = a(v w) for all v w isin Vk


1 v = u(i) +RJ(bminus AJu(i))

2 w = v + LJ VcycleJminus1(0 LtJ (bminus AJv))

3 u(i+1) = w +RtJ(bminus AJw)


Jay Gopalakrishnan

Abstract theory

THEOREM If Kk equiv I minusRkAk satisfies


[


]

forallv isin Vk

then|||Ek|||a le δ equiv

C

C + 1

PROOF a(Ekv v) =

= |||(I minus Pkminus1)Kku|||2a + a(Ekminus1Pkminus1Kku Pkminus1Kku)

le |||(I minus Pkminus1)Kkv|||2a + δ|||Pkminus1Kkv|||

2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]

|||(I minus Pkminus1)Kkv|||2a + δ|||Pkminus1Kkv|||

2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)|||(I minus Pkminus1)Kkv|||2a + δ|||Kkv|||

2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a

Result follows since (1minus δ)C = δDepartment of Mathematics [Slide 23 of 36]

Jay Gopalakrishnan

Applying the theorem


[


]

forallv isin Vk

We verified the condition of the abstract theorem in the case ofthe Richardson smoother Similar arguments prove it for theGauszlig-Seidel iteration and a scaled Jacobi iteration providedwe have the following For every f isin Vk the exact solution of

a(φ v) = 〈f v〉 forallv isin V


|||φminus φk|||a le Chkf

This is the REGULARITY amp APPROXIMATION property[Bramble amp Pasciak 1987]


Jay Gopalakrishnan



[


]

forallv isin Vk





It holds for our Eg 1 on domains admitting full regularity

|||φminus φk|||a le Chk|φ|H2(Ω) |φ|H2(Ω) le CfL2(Ω)Department of Mathematics [Slide 24 of 36]

Jay Gopalakrishnan



[


]

forallv isin Vk





The theory can be extended to situations with less than fullregularity [Bramble amp Pasciak 1987]


Jay Gopalakrishnan

Eg 2 A degenerate elliptic eq

minuspartx(ε(x y)partxu)minus party(b(x y)partyu) = f on Ω

u = 0 on partΩ

Here 0 lt b0 le b(x y) but 0 lt ε(x y) can be degenerateHence the problem is not uniformly elliptic

Therefore the intuition with which we built the multigridalgorithm previously seems to fail

While higher eigenfunctions become more oscillatory forelliptic operators for degenerate elliptic operators even thelower eigenfunctions can be oscillatory in which case thelower eigenfunction components cannot be resolved by thecoarse grid


Jay Gopalakrishnan



u = 0 on partΩ


Here are the 1st 5th and 10th eigenfunctions of adiscretization of this operator (when ε = 0001 b = 1)


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Eg 2 Failure of smootherAs expected the performance of the standard Gauszlig-Seideliteration as a smoother is not satisfactory for this example

We pick a random vector apply the Gauszlig-Seidelreducer a few times and observe the resultscomputationally (Click to see code)


Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy


While the resulting vector is smooth in the y-direction it is os-

cillatory in the x-direction


Jay Gopalakrishnan

Eg 2 Modifying the smootherRecall


[


]

︸︷︷︸

forallv isin Vk



Thus for multigrid to work the undamped components aftersmoothing must be representable on coarser grids

Therefore the smoother must damp not only the highereigenfunctions but also some of the lower ones like

as these are not well represented on coarse grid


Jay Gopalakrishnan

Eg 2 Modifying the smootherTherefore the smoother must damp not only the highereigenfunctions but also some of the lower ones like


minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1


Such functions are damped if we use a block Jacobi iterationwith each block consisting of unknowns along y-lines rarr


Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy

The 10th eigenfunction Result after 6 point Gauszlig-Seidel iterations

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy

larrminus Result after one iteration of the

line Jacobi smoother


(Now R is a block diagonal matrix in an

appropriate ordering) Department of Mathematics [Slide 28 of 36]

Jay Gopalakrishnan

Eg 2 TheoryThe V-cycle with line smoothers were analyzed by[Bramble amp Zhang 2000] Here are their results If u solves


u = 0 on partΩ thenLEMMA (REGULARITY)

ε12partxxu2L2(Ω)+partyyu

2L2(Ω)+partxyu

2L2(Ω) le Cεminus12f2L2(Ω)

LEMMA (APPROXIMATION) There is a uk isin Vk such that

|||uminus uk|||ale Chk

[


2L2(Ω)+partxyu

2L2(Ω)

]

Hence |||(I minus Pkminus1)Kkv|||2a le C

[


]


Jay Gopalakrishnan

Eg 3 Axisymmetric Laplace eq

Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D

Significant computational savings (3D to 2D domain)

But numerical analyses face difficulties due to Γ0


Jay Gopalakrishnan


Dirichlet problem on Ω Reduced problem on D

minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0

For smooth functions φ since partrφ is an even function of r

partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0

Weak formulation Find u isin V such thatint

D

r(partru)(partrv) + r(partzu)(partzv) drdz =

int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V equiv w isin L2r(D) partrw partzw isin L

2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan

Eg 3 BackgroundIn view of the fact that line smoothings were necessary for thedegenerate problem of Eg 2 and because of the similarpresence of the singular coefficients in the current example ofthe axisymmetric equation

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f

multigrid algorithms with line smoothing iterations andmultilevel grids with semicoarsening have usually beensuggested for this problem until recently

We will prove that neither is necessary


Jay Gopalakrishnan

Eg 3 Bilinear elements

Mesh D by square elements Vh = bilinear FE subspace of VExact solution u isin V

ar(u v) = (f v)r forallv isin V

Finite element approximation uh isin Vh

ar(uh vh) = (f vh)r forallvh isin Vh

Error analysis

|uminus uh|H1r(D) le inf

vhisinVh

|uminus vh|H1r(D) (standard)

le |uminus Πhu|H1r(D) (non-standard Πh)

le Ch|u|H2r (D) le ChfL2

r(D) (wtd regularity)


Jay Gopalakrishnan

Eg 3 Convergence of V-cycle

We apply the abstract theory with form a(middot middot) = ar(middot middot) baseinnerproduct 〈middot middot〉 = (middot middot)r and use the REGULARITY (whichholds on convex domains) and APPROXIMATION property to get


[


]

forallv isin Vk

THEOREM If Ω is convex the multigrid V-cycle with pointGauszlig-Seidel or Jacobi smoother applied to the bilinear finiteelement discretization of the axisymmetric Laplace equationconverges at a rate independent of mesh sizes [G amp Pasciak 2005]

Thus not all problems with degenerate coefficients requiresemicoarsening and line relaxations


Jay Gopalakrishnan

Eg 3 Numerical results

We numerically estimated the operator norm |||EJ |||a by using afew iterations of the Lanczos method

J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017

These values show that the practical convergence rates in|||middot|||andashnorm of the V-cycle are excellent

Ω = (minus1 1)2

Bilinear finite elements used

Meshes are obtained by dividing Ω into ntimes n square with n = 2k


Jay Gopalakrishnan

ConclusionWe saw how one builds multigrid algorithms using thebasic ingredients of smoothing prolongation andmultilevel recursion

We discussed the theory of multigrid convergence andsaw how elliptic regularity and approximation propertiesplayed a crucial role

We applied the multigrid paradigm to the followingexamples

Eg 1 Laplacersquos equation

Eg 2 A degenerate anisotropic elliptic problem

Eg 3 Axisymmetric Poisson equation

In Part II of this course we will see newer multigrid theoriesand advanced applications in electromagnetics


Why multigrid

Structure of multigrid algorithms

The multigrid idea




Eg 1 Prolongation

Elliptic eigenfunctions

Richardson smoother

Eg 1 The algorithm


Multigrid cycles

Braess-Hackbusch theorem

Proof

Proof

Regularity amp Approximation

Practical smoothers



The smoothing condition

Abstract V-cycle

Abstract theory



Eg 2 Failure of smoother

Eg 2 Modifying the smoother

Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Why multigrid

Multigrid techniques give algorithms that solve sparse linearsystems

Ax = b

of N unknowns with O(N) work and storage for large classesof problems It applies typically to systems arising fromdiscretization of partial differential equations



SOR δ lt 1minus Ch





Jay Gopalakrishnan

Why multigrid












Jay Gopalakrishnan

Why multigrid







Source




Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Why multigrid












Jay Gopalakrishnan

Why multigrid







Source




Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Why multigrid







Source




Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



A 2D example

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan








Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined




u(i+1) = MgJ(u(i) b) i = 0 1 2



Idea





Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined





Jr



Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

The multigrid idea

k = 1 k = 2

k = J

Highlyrefined








Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined







Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

Prolong2

Restrict1




2 Correction



Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


k = 1 k = 2

k = J

Highlyrefined

L2

Lt2




2 Correction



Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


FEM




Weakform




u = 0 on partΩ


Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


FEM




Operator


Ahuh = fh





Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



the kth level mesh



Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



Ω


V1 V2 VJ equiv Vh



2 Correction



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk


Ω



Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Eg 1 Prolongation




Vkminus1 rarr Vk



2 Correction



Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan





minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Richardson smoother




(i))




(i))



(i)






=rArr (I minus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max




Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


λ(k)max

)ψn


+ ellipticity


Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Richardson smoother




(i))





=rArr (Iminus1

λ(k)max

Ak)e = c1(1minusλ1

λ(k)max


(k)max

λ(k)max

)ψn



Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




2 Correction



Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Eg 1 The algorithm


1 Smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction




Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



1 Pre-smoothing

v = u(i) +1

λ(k)max

(bminus AJu(i))

2 Correction


3 Post-smoothing

u(i+1) = w +1

λ(k)max

(bminus AJw)


Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


-cycle

FMG schedule

F-cycle

W-cycle

V-cycle

hJ

hJminus1

h1

hJ

hJminus1

h1


Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan






|||Ek|||a le δ


Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




[


]

forallv isin Vk






[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




[


]

︸︷︷︸

forallv isin Vk







[


]

forallv isin Vk


Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




[


]

forallv isin Vk





Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic(AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

λ(k)max

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

Chminus2k

(1

λ(k)max

AkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

Chminus2k

(

w minusKkwAkw

)

radic

Chminus2k

[

a(ww)minus a(Kkww)

]


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



radic

a(ww)minus a(Kkww)


Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]



︸︷︷︸

radic

a(ww)minus a(Kkww)



Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Proof




radic

Chminus2k

[

a(ww)minus a(Kkww)

]


a(ww)minus a(Kkww)



[


]

forallv isin Vk


Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan










Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Practical smoothers







Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan










Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan










Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

xy



Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan





[


]

forallv isin Vk




Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan











Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le

[

(1minus δ) + δ

]


2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =



2a

(1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Abstract theory



[


]

forallv isin Vk


C

C + 1

PROOF a(Ekv v) =


le (1minus δ)C

[


]

+ δ|||Kkv|||2a


Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



[


]

forallv isin Vk







Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



u = 0 on partΩ





Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



u = 0 on partΩ




minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


minus08

minus06

minus04

minus02

0

02

04

06

08

1


Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


minus1

minus05

0

05

1

minus1

minus05

0

05

10

02

04

06

08

1

xy minus1

minus05

0

05

1

minus1

minus05

0

05

10

01

02

03

04

05

06

07

08

xy





Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



[


]

︸︷︷︸

forallv isin Vk







Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1



Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus08

minus06

minus04

minus02

0

02

04

06

08

1




Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan

Eg 2 Line smoother

minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy


minus1

minus05

0

05

1

minus1

minus05

0

05

1minus01

minus005

0

005

01

xy






Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan





2L2(Ω)+partxyu




[


2L2(Ω)+partxyu

2L2(Ω)

]


[


]


Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


Γ0

Γ1

DΩ 0

z

r

Ω minusrarr D




Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

on Γ0


partrφ|Γ0= 0


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz

forallv isin V


Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



minus∆U = f on Ω

U = 0 on partΩ

(f is axisymmetric)

minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f on D

u = 0 on Γ1

partru = 0 on Γ0


D


int

D

fv r drdz


2r(D)

︸︷︷︸

H1r (D)

w|Γ1= 0


Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan


minus1

r

part

partr

(

rpartu

partr

)

minuspart2u

partz2= f




Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan






Error analysis


vhisinVh






Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan




[


]

forallv isin Vk




Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan



J 2 3 4 5 6 7 8 9 10

|||EJ |||a 012 016 017 017 017 017 017 017 017


Ω = (minus1 1)2




Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan









Why multigrid


The multigrid idea




Eg 1 Prolongation


Richardson smoother

Eg 1 The algorithm


Multigrid cycles


Proof

Proof


Practical smoothers




Abstract V-cycle

Abstract theory





Eg 2 Line smoother

Eg 2 Theory



Eg 3 Background




Conclusion

Jay Gopalakrishnan University of Florida Montreal ...web.pdx.edu/~gjay/talks/montrealI.pdf · Jay...

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Transcript of Jay Gopalakrishnan University of Florida Montreal ...web.pdx.edu/~gjay/talks/montrealI.pdf · Jay...