Unified Hybridization Of Discontinuous Galerkin, Mixed ...€¦ · To CG or to HDG: A Comparative...

Unified Hybridization Of Discontinuous Galerkin, Mixed,

And Continuous Galerkin Methods For Second Order

Elliptic Problems.

Stefan Girke

WWU MünsterInstitut für Numerische und Angewandte Mathematik

10th of January, 2011

Stefan Girke (WWU Münster Institut für Numerische und Angewandte Mathematik)Hybridization 10th of January, 2011 1 / 43

Inhalt

1 introduction (an example)


Inhalt


2 the framework


Inhalt


2 the framework

3 examples for hybridizable methods


Inhalt


2 the framework


4 Other novel methodshybridizable methods well suited for adaptivityThe RT-method on meshes with hanging nodes


Inhalt

literature

D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini.

Unified analysis of discontinuous Galerkin methods for elliptic problems.SIAM journal on numerical analysis, 39(5):1749–1779, 2002.

B. Cockburn and J. Gopalakrishnan.

A characterization of hybridized mixed methods for second order elliptic problems.SIAM Journal on Numerical Analysis, 42(1):283–301, 2005.

B. Cockburn, J. Gopalakrishnan, and R. Lazarov.

Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods for second order ellipticproblems.SIAM J. Numer. Anal, 47(2):1319–1365, 2009.

R.M. Kirby, S.J. Sherwin, and B. Cockburn.

To CG or to HDG: A Comparative Study.

M. Vohraĺık, J. Maryska, and O. Severýn.

Mixed and nonconforming finite element methods on a system of polygons.Applied Numerical Mathematics, 57(2):176 – 193, 2007.

OC Zienkiewicz.

Displacement and equilibrium models in the finite element method by B. Fraeijs de Veubeke, Chapter 9, Pages 145-197of Stress Analysis, Edited by OC Zienkiewicz and GS Holister, Published by John Wiley & Sons, 1965.International Journal for Numerical Methods in Engineering, 52(3):287–342, 2001.


introduction (an example)


2 the framework





Second order elliptic boundary value problem

Search solution u for

∇ · (a∇u) + du = f in Ω ⊂ Rn,u = g on ∂Ω,

where

a(x) is a bounded symmetric positive definite matrix-valued function,






where


d(x) is a bounded nonnegative function.






where


d(x) is a bounded nonnegative function.

This can be rewritten

mixed form

Search solution (q, u) for

q+ a∇u = 0 in Ω,

∇ · q+ du = f in Ω,

u = g on ∂Ω.



We’re choosing a triangulation of Ω, Th

Th



We’re choosing a triangulation of Ω, Th

Th

and using Raviart-Thomas elements Pk(K )n + xPk(K )

RT0 RT1 RT2

b

b

b b



Notation

E◦h interior edges, E∂h boundary faces of Ω,



Notation


Eh = E◦h ∪ E

∂h ,



Notation


Eh = E◦h ∪ E

∂h ,

(u, v)D :=∫Duvdx for u, v ∈ L2(D), D ⊂ Rn,



Notation


Eh = E◦h ∪ E

∂h ,


〈u, v〉D :=∫Duvdx for u, v ∈ L2(D), D ⊂ Rn−1,



Notation


Eh = E◦h ∪ E

∂h ,



(u, v)Th =∑

K∈Th

(v ,w)K for u, v on Ω,



Notation


Eh = E◦h ∪ E

∂h ,



(u, v)Th =∑

K∈Th


〈µ, λ〉E =∑e∈E

〈µ, λ〉e for µ, λ on E ⊂ Eh.



Notation


Eh = E◦h ∪ E

∂h ,



(u, v)Th =∑

K∈Th


〈µ, λ〉E =∑e∈E

〈µ, λ〉e for µ, λ on E ⊂ Eh.

H(div,Ω) := {v ∈ L2(Ω)n | ∇ · v ∈ L2(Ω)}



RT finite element approximation

The approximation (qh, uh) is sought in the finite element spaceVRTh ×W

RTh given by

VRTh = {v ∈ H(div,Ω) | v|K ∈ Pk(K )n + xPk(K ) ∀K ∈ Th},

W RTh = {w ∈ L2(Ω) | w |K ∈ P

k(K ) ∀K ∈ Th}.

and is defined by requiring that,



RT finite element approximation

The approximation (qh, uh) is sought in the finite element spaceVRTh ×W

RTh given by

VRTh = {v ∈ H(div,Ω) | v|K ∈ Pk(K )n + xPk(K ) ∀K ∈ Th},

W RTh = {w ∈ L2(Ω) | w |K ∈ P

k(K ) ∀K ∈ Th}.


(cqh, v)Ω − (uh,∇ · v)Ω = −〈g , v · n〉∂Ω ∀v ∈ VRTh ,

(w ,∇ · qh)Ω − (duh,w)Ω = (f ,w)Ω ∀w ∈ WRTh ,

with c = a−1 for each element K ∈ Th



This yields to a matrix equation of the form:

(A −BtB D )(QU) = (GH) ,where Q and U are vectors of coefficients of qh and uh with respect totheir corresponding finite element basis.



problem

System is not positive definite.



problem


solutions

using expensive solver,



problem


solutions


using a positive definite system by elimination of Q:solve equation

(BA−1Bt + D)U = F+ BA−1Gfor U. Required inversion A−1 is difficult to compute and a full matrix.



problem


solutions


using a positive definite system by elimination of Q:solve equation

(BA−1Bt + D)U = F+ BA−1Gfor U. Required inversion A−1 is difficult to compute and a full matrix.hybridization:Introduce new unkowns λh (Lagrangian multipliers) and relax thecontinuity constraints between element interfaces for qh.



Definition (hybridized RT finite element approximation)

The approximation (qh, uh, λh) is sought in the finite element space

V̂hRT

× ŴhRT

× M̂◦hRT

given by

V̂hRT

= {v ∈ L2(Ω)n | v|K ∈ Pk(K )n + xPk(K ) ∀K ∈ Th},

ŴhRT

= {w ∈ L2(Ω) | w |K ∈ Pk(K ) ∀K ∈ Th},

M̂◦hRT

= {µ ∈ L2(E◦h ) | µ|e ∈ Pk(e) ∀e ∈ E◦h}



Definition (hybridized RT finite element approximation)

The approximation (qh, uh, λh) is sought in the finite element space

V̂hRT

× ŴhRT

× M̂◦hRT

given by

V̂hRT

= {v ∈ L2(Ω)n | v|K ∈ Pk(K )n + xPk(K ) ∀K ∈ Th},

ŴhRT

= {w ∈ L2(Ω) | w |K ∈ Pk(K ) ∀K ∈ Th},

M̂◦hRT

= {µ ∈ L2(E◦h ) | µ|e ∈ Pk(e) ∀e ∈ E◦h}


(cqh, v)Ω − (uh,∇ · v)Ω + 〈λh, JvK〉E◦h = −〈g , v · n〉∂Ω ∀v ∈ V̂hRT

,

(w ,∇ · qh)Ω − (duh,w)Ω = (f ,w)Ω ∀w ∈ ŴhRT

,

〈µ, JqhK〉E◦h = 0 ∀µ ∈ M̂◦h

RT.



This yields to a matrix of the form

A −B t −C t

B D 0C 0 0

QUΛ

=

GF0

,

where Λ is the vector of dofs associated to the multiplier λh.




A −B t −C t

B D 0C 0 0

QUΛ

=

GF0

,

where Λ is the vector of dofs associated to the multiplier λh.New vectors of dofs Q and U define the same approximation (qh, uh) asthe original method.




A −B t −C t

B D 0C 0 0

QUΛ

=

GF0

,


Both Q and U can be easily eliminated




A −B t −C t

B D 0C 0 0

QUΛ

=

GF0

,


Both Q and U can be easily eliminatedEΛ = H,E =CA−1(A − B t(BA−1B t + D)−1B)A−1C t ,H =− CA−1(A− B t(BA−1B t + D)−1B)A−1G− CA−1B t(BA−1B t + D)−1F .



advantages of hybridization

A is now a block diagonal matrix,




A is now a block diagonal matrix,E is symmetric positive definite,




A is now a block diagonal matrix,E is symmetric positive definite,number of dofs is remarkably smaller,





once Λ has been obtained, both Q and U can be computed efficientlyelement by element,





once Λ has been obtained, both Q and U can be computed efficientlyelement by element,

the multiplier λh can be used to improve the approximation to u bymeans of local post-processing.



history of hybridization

year event

1965 first hybridization of finite elements for solving equationsof linear elasticity (often: static condensation)/“implementation trick“[6]

1985 proof: hybrid variable contains extra informationabout the exact solution (⇒ local post-processing)[5]

2004 hybridized RT and BDM methods of arbitrary order[2]2005 extended to finite element methods for

stationary Stokes equation (DG, mixed)2009 unifying framework[3]2010 comparison CG-H vs. LDG-H[4]


the framework


2 the framework




the framework

Unified framework provides approximations for

1 (q, u) in the interior of the elements K ∈ Th, (qh, uh),

2 u on the interior border of the elements, λh.


the framework

Definition (local solvers)

For any single valued function m ∈ L2(∂K ), the functions (Qm,Um) arethe solutions of

cQm+∇Um = 0 on K ,

∇ ·Qm+ dUm = 0 on K ,

Um = m on ∂K


the framework

Definition (local solvers)

For any single valued function m ∈ L2(∂K ), the functions (Qm,Um) arethe solutions of

cQm+∇Um = 0 on K ,

∇ ·Qm+ dUm = 0 on K ,

Um = m on ∂K

and for any single valued function f ∈ L2(K ), the functions (Qf ,Uf ) arethe solutions of

cQf +∇Uf = 0 on K ,

∇ ·Qf + dUf = f on K ,

Uf = 0 on ∂K .


the framework

With λu g= +and the linearity of the problem we have that

(q, u) = (Qλ+Qg +Qf ,Uλ+ Ug + Uf ).


the framework

With λu g= +and the linearity of the problem we have that

(q, u) = (Qλ+Qg +Qf ,Uλ+ Ug + Uf ).

The above property only holds iff

Definition (transmission condition)

JQλ+Qg +Qf K = 0

This completely characterizes λ.


the framework

Definition (discrete local solvers)

(Qm,Um) and (Qf ,Uf ) are the discrete versions of the local solvers.


the framework

Definition (discrete local solvers)

(Qm,Um) and (Qf ,Uf ) are the discrete versions of the local solvers.

Then the discrete solution can be written as

(qh, uh) = (Qλh +Qgh +Qf ,Uλh + Ugh + Uf ),

where λh ∈ M◦h and gh ∈ M

∂h are approximations to the values of u.

uh λh ∈ M◦h gh ∈ M

∂h


the framework

Definition (discrete version of transmission condition (weak))

ah(λh, µ) = bh(µ) ∀µ ∈ M◦h

is the discrete version of the transmission condition.


the framework

Notation

V(K ) polynomial space in which q is approximated,


the framework

Notation


W (K ) polynomial space in which u is approximated,


the framework

Notation



Vh = {v | v|K ∈ V(K )},


the framework

Notation



Vh = {v | v|K ∈ V(K )},

Wh = {w | w |K ∈ W (K )}.


the framework

Notation



Vh = {v | v|K ∈ V(K )},

Wh = {w | w |K ∈ W (K )}.

In general: A function h on K is double-valued, each branch is denoted byhK+ or hK− .


the framework

To define a hybridizable method, we had to define

the discrete local solvers (Qm,Um) and (Qf ,Uf ),


the framework



the transmission condition determining λh, ah and bh,


the framework




the finite element spaces M◦h and M∂h ,


the framework





the finite element spaces V(K ) and W (K ),


the framework






the trace spaces Q̂m and Q̂f (used in discrete local solvers).


the framework






the trace spaces Q̂m and Q̂f (used in discrete local solvers).

Finally, think about existence and uniqueness of λh.


the framework

Definition (trace space M◦h )

M◦h := {µ ∈ M∂h | µ = 0 on ∂Ω}.


the framework

Definition (discrete local solver)

1 maps m ∈ M∂h to the function (Qm,Um):

(cQm, v)K − (Um,∇ · v)K = −〈m, v · n〉∂K ∀v ∈ V(K ),

−(∇w ,Qm)K + 〈w , Q̂m · n〉∂K + (dUm,w)K = 0 ∀w ∈ W (K ).


the framework

Definition (discrete local solver)

1 maps m ∈ M∂h to the function (Qm,Um):

(cQm, v)K − (Um,∇ · v)K = −〈m, v · n〉∂K ∀v ∈ V(K ),

−(∇w ,Qm)K + 〈w , Q̂m · n〉∂K + (dUm,w)K = 0 ∀w ∈ W (K ).

2 maps f ∈ L2(Ω) to the pair (Qf ,Uf ):

(cQf , v)K − (Uf ,∇ · v)K = 0 ∀v ∈ V(K ),

−(∇w ,Qf )K + 〈w , Q̂f · n〉∂K + (dUf ,w)K = (f ,w)K ∀w ∈ W (K ).


the framework

Definition (characterization of λh)

ah(η, µ) = (cQη,Qµ)Th + (dUη,Uµ)Th + 〈1, J(Uµ− µ)(Q̂η −Qη)K〉Eh ,

bh(µ) = 〈gh, JQ̂µK〉Eh + (f ,Uµ)Th − 〈1, J(Uµ− µ)(Q̂f −Qf )K〉Eh

+ 〈1, JUf (Q̂µ−Qµ)K〉Eh

− 〈1, J(Uµ− µ)(Q̂gh −Qgh)K〉Eh

+ 〈1, J(Ugh − g)(Q̂µ−Qµ)K〉Eh .


the framework

Theorem (existence and uniqueness of λh)

If the three assumptions

1

2

3

are fulfilled, then there is a unique solution λh of the weak formulation.


the framework



1 existence and uniqueness of the local solvers,

2

3



the framework




2 positive semidefiniteness of the local solvers and

3



the framework




2 positive semidefiniteness of the local solvers and

3 the “gluing condition” (If µ ∈ M∂h , then for every interior face thereexists a branch ∂K with P∂Kµ = µ.)



examples for hybridizable methods


2 the framework





general assumptions

same local solvers in every element K ,



general assumptions


conforming simplicial triangulation.



general assumptions


conforming simplicial triangulation.

Mch,k := {µ ∈ C(Eh) | µ|e ∈ Pk(e) for all faces e ∈ Eh}

Mh,k := {µ ∈ L2(Eh) | µ|e ∈ Pk(e) for all faces e ∈ E

◦h}



RT-H (Raviart Thomas):

V(K )×W (K ) Pk(K )n + xPk(K )× Pk(K )

M∂h Mh,kQ̂m Qm

Q̂f Qf

ah(η, µ) (cQη,Qµ)Th + (dUη,Uµ)Thbh(µ) 〈gh,Qµ · n〉∂Ω + (f ,Uµ)ThConservativity strong



BDM-H (Brezzi Douglas Marini):

V(K )×W (K ) Pk(K )n × Pk−1(K )

M∂h Mh,kQ̂m Qm

Q̂f Qf

ah(η, µ) (cQη,Qµ)Th + (dUη,Uµ)Thbh(µ) 〈gh,Qµ · n〉∂Ω + (f ,Uµ)ThConservativity strong



LDG-H (Local Discontinuous Galerkin):V(K )×W (K ) 1.) Pk(K )

n × Pk−1(K )2.) Pk(K )

n × Pk(K )3.) Pk−1(K )

n × Pk(K )

M∂h Mh,kQ̂m Qm+ τK (Um−m)n

Q̂f Qf + τK (Uf )n

ah(η, µ) (cQη,Qµ)Th + (dUη,Uµ)Th+〈1, J(Uµ− µ)(τK (Uη − η)n)K〉Eh

bh(µ) 〈gh,Qµ · n+ τKUµ〉∂Ω + (f ,Uµ)ThConservativity strongThe stabilization parameter τK is nonnegative constant on each face in Eh,double-valued on E◦h .



CG-H (Continuous Galerkin):V(K )×W (K ) Pk−1(K )

n × Pk(K )

M∂h Mch,k

Q̂m a new unkown variable

Q̂f a new unkown variable

ah(η, µ) (a∇Uη,∇Uµ)Th + (dUη,Uµ)Thbh(µ) 〈gh, JQ̂µK〉Eh + (f ,Uµ)ThConservativity weak

Assume that a(x) is a constant on each element.CG-H is an LDG-H method with τ → ∞ .



Are all methods hybridizable?



Are all methods hybridizable?No.



Are all methods hybridizable?No.

Example (DG-methods)

Remember the lecture “Scientific Computing” or see [1].A hybrizable method has to be single valued for ûh (= λh).

Method ûhBassi-Rebay {uh}Brezzi et al. {uh}LDG {uh} − β · JuhKIP {uh}Bassi et al. {uh}Baumann-Oden {uh}+ nK · JuhKNIPG {uh}+ nK · JuhKBabuska-Zlamal (uh|K )|∂KBrezzi et al. (uh|K )|∂K


Other novel methods


2 the framework




Other novel methods hybridizable methods well suited for adaptivity


2 the framework





hybridizable methods well suited for adaptivity:V(K )×W (K ) 1.) Pk(K)(K )

n + xPk(K) × Pk(K)(K )

2.) Pk(K)(K )n × Pk(K)−1(K )

3.) Pk(K)(K )n × Pk(K)(K )

4.) Pk(K)−1(K )n × Pk(K)(K )

M∂h {µ | µ|e ∈ Mh,k(e) ∀e ∈ E◦h}

∩{µ | µ|∂K ∈ C({x ∈ ∂K | τK (x) = ∞})}

Q̂m Qm+ τK (Um−m)n

Q̂f Qf + τK (Uf )n

ah(η, µ) depending on V(K ),W (K )bh(µ) depending on V(K ),W (K )with τK 6= 0 for3.) (on at least one face) and4.) (for alle faces).



For e = ∂K+ ∩ ∂K−, we set

k(e) :=

max{k(K+), k(K−)}, if τ+ < ∞ and τ− < ∞

k(K±), if τ± = ∞ and τ∓ < ∞

min{k(K+), k(K−)}, if τ+ = ∞ and τ− = ∞.



assumption

[0,∞] ∋ τK

τK ∈ (0,∞)

K

interior face E◦h

face of Th corresponding to LDG-H

depending on the selected method



Main features of this class of methods:

1 variable degree approximation spaces on conforming meshes (k(K )),





2 automatical coupling of different methods on conforming meshes (τKdetermines method),





2 automatical coupling of different methods on conforming meshes (τKdetermines method),

3 mortaring capabilities for nonconforming meshes (choice of τ forhanging nodes).


Other novel methods The RT-method on meshes with hanging nodes


2 the framework





Considere the case of variable degree RT-H method with τ ≡ 0everywhere (assumption in last method is violated).




Mesh is locally refined into four congruent triangles.




Mesh is locally refined into four congruent triangles.

Idea: Impose special conditions on the meshes and link the definitionof k(K ) to the structure of the mesh.



k(K1)k(K2)

k(K4)

k(K3)

"≥"

"≥"

max{k(K

3),k(K

4)}

max{k(K2), k(K4)}

arbitrarily



Thanks for your attention!Stefan Girke (WWU Münster Institut für Numerische und Angewandte Mathematik)Hybridization 10th of January, 2011 43 / 43

Inhaltintroduction (an example)the frameworkexamples for hybridizable methodsOther novel methodshybridizable methods well suited for adaptivityThe RT-method on meshes with hanging nodes

Unified Hybridization Of Discontinuous Galerkin, Mixed ...€¦ · To CG or to HDG: A Comparative...

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