Streamline Poincaré-Friedrichs inequality and ... · Streamline Poincar e-Friedrichs inequality...

Convection-diffusion equations and preconditioning Poincare-Friedrichs inequalities Robust preconditioning estimates

Streamline Poincare-Friedrichs inequality andpreconditioning estimates

Janos [email protected]

Department of Applied Analysis and Comp. Math. & Numnet Research GroupELTE University, Budapest, Hungary

3 March 2014

February 25, 2014

J. Karatson Department of Applied Analysis, ELTE

Streamline diffusion preconditioning

Outline of the talk

Convection-diffusion equations and preconditioning

PreliminariesThe diffusion-dominated caseThe convection-dominated case

Streamline Poincare-Friedrichs inequality

Robust preconditioning estimates

Preliminaries

Convection-diffusion equations

A simple Dirichlet problem:−ε∆u + w · ∇u = g

u|∂Ω = 0.(1)

Assumptions:

(i) Ω ⊂ Rn is a polyhedral domain.

(ii) w ∈ C 1(Ω, Rn), divw = 0.

(iii) g ∈ L2(Ω).

Preliminaries

u|∂Ω = 0.(1)

Assumptions:

(i) Ω ⊂ Rn is a polyhedral domain.

(ii) w ∈ C 1(Ω, Rn), divw = 0.

(iii) g ∈ L2(Ω).

Preliminaries

Remark. More general mixed BVPs:−ε∆u + w · ∇u + qu = g

u|ΓD= 0, ∂u

∂ν + βu|ΓN= 0.

Assumptions:

(i) Ω ⊂ Rn is a polyhedral domain; ∂Ω = ΓD ∪ ΓN .

(ii) w ∈ C 1(Ω, Rn), q ∈ L∞(Ω), β ∈ L∞(ΓN).

(iii) q − 12 divw ≥ 0 in Ω, w · ν ≥ 0 on ΓN .

(iv) g ∈ L2(Ω).

Preliminaries

Remark. More general mixed BVPs:−ε∆u + w · ∇u + qu = g

u|ΓD= 0, ∂u

∂ν + βu|ΓN= 0.

Assumptions:

(i) Ω ⊂ Rn is a polyhedral domain; ∂Ω = ΓD ∪ ΓN .

(ii) w ∈ C 1(Ω, Rn), q ∈ L∞(Ω), β ∈ L∞(ΓN).

(iii) q − 12 divw ≥ 0 in Ω, w · ν ≥ 0 on ΓN .

(iv) g ∈ L2(Ω).

Preliminaries

The Dirichlet problem: weak solution.

Bilinear form on H10 (Ω)× H1

0 (Ω):

a(u, v) :=

(ε∇u · ∇v + (w · ∇u)v

Right-hand side functional: `v :=

Weak formulation:

a(u, v) = `v (∀v ∈ H10 (Ω)).

Preliminaries

Bilinear form on H10 (Ω)× H1

0 (Ω):

a(u, v) :=

(ε∇u · ∇v + (w · ∇u)v

Right-hand side functional: `v :=

Weak formulation:

a(u, v) = `v (∀v ∈ H10 (Ω)).

Preliminaries

Coercivity and boundedness of the bilinear form

⇒ solvability.

Preliminaries

Finite element solution

Type of FEM: depending on the dominating term.

The diffusion-dominated case: ε = O(|w|) → standard FEM

Preliminaries

Finite element solution

Type of FEM: depending on the dominating term.

The diffusion-dominated case: ε = O(|w|) → standard FEM

Preliminaries

Finite element solution: diffusion-dominated case

Standard FEM

FEM subspace: Vh ⊂ H10 (Ω)

Find uh ∈ Vh:

a(uh, vh) = `vh (∀vh ∈ Vh).

Coefficients for uh =∑

ciϕi :

linear algebraic systemAhc = bh. (LAER)

Nonsymmetric but positive definite.

Preliminaries

Standard FEM

Find uh ∈ Vh:

ciϕi :

Preliminaries

Standard FEM

Find uh ∈ Vh:

ciϕi :

Iterative solution

Iteration and preconditioning

Solution of the LAER - iteration and preconditioning.

Iteration: GCG-LS or CGN.

Convergence for a LAER Ax = b:

depends on the bounds

λ(A) := min〈Ax , x〉‖x‖2

Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖

Iterative solution

λ(A) := min〈Ax , x〉‖x‖2

Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖

Iterative solution

λ(A) := min〈Ax , x〉‖x‖2

Λ(A) := ‖A‖ = max〈Ax , y〉‖x‖‖y‖

Iterative solution

Solution of the LAER - iteration.

Convergence: let λ = λ(A), Λ = Λ(A), then

GCG-LS:‖rk‖‖r0‖

1−(λ

)2)k/2

CGN:‖rk‖‖r0‖

≤ 2(Λ− λ

Λ + λ

)k⇒ convergence depends on k(A) :=

Problem: k(Ah) = O(h−2)→∞ as h→ 0.

Iterative solution

1−(λ

)2)k/2

≤ 2(Λ− λ

Λ + λ

Iterative solution

1−(λ

)2)k/2

≤ 2(Λ− λ

Λ + λ

Iterative solution

1−(λ

)2)k/2

≤ 2(Λ− λ

Λ + λ

Iterative solution

Solution of the LAER - preconditioning.

Ax = b → S−1Ax = S−1b

Optimal case: O(N) operations for solving with S ∈ RN×N

Iterative solution

Solution of the LAER - preconditioning.

Ax = b → S−1Ax = S−1b

Optimal case: O(N) operations for solving with S ∈ RN×N

Iterative solution

Operator preconditioning

Return to the FEM solution of the convection-diffusion problem.

Linear algebraic system:

Ahc = bh, (LAER)

where(Ah)ij = a(ϕj , ϕi ).

Preconditioning: Sh =?

Iterative solution

Return to the FEM solution of the convection-diffusion problem.

Linear algebraic system:

Ahc = bh, (LAER)

where(Ah)ij = a(ϕj , ϕi ).

Preconditioning: Sh =?

Iterative solution

Optimal preconditioner:

stiffness matrix for a symmetric elliptic problem,

(Sh)ij = b(ϕj , ϕi ) = b(ϕi , ϕj)

→ optimal O(N) solvers are available (multigrid, multilevel)

(or quasi-optimal, O(N log N) like FFT)

E.g.: b(u, v) =

∫Ω∇u · ∇v

(−∆u)v)

→ induces the H10 -norm: b(u, u) =

∫Ω |∇u|2 =: |u|21

Iterative solution

Optimal preconditioner:

stiffness matrix for a symmetric elliptic problem,

(Sh)ij = b(ϕj , ϕi ) = b(ϕi , ϕj)

→ optimal O(N) solvers are available (multigrid, multilevel)

(or quasi-optimal, O(N log N) like FFT)

E.g.: b(u, v) =

∫Ω∇u · ∇v

(−∆u)v)

→ induces the H10 -norm: b(u, u) =

∫Ω |∇u|2 =: |u|21

Iterative solution

Conditioning properties.

Connection between a and b:

coercivity and boundedness of a w.r.t. the | . |1-norm ⇒

|a(u, v)| ≤ M√

b(u, u)b(v , v), a(u, u) ≥ m b(u, u).

Iterative solution

Connection between a and Ah, respectively b and Bh:

if uh =∑

ciϕi and vh =∑

diϕi , then

Ahc · d = a(uh, vh), Bhc · d = b(uh, vh).

In particular: |c|2Bh:= Bhc · c = b(uh, uh) = |uh|21.

Iterative solution

if uh =∑

ciϕi and vh =∑

diϕi , then

Iterative solution

if uh =∑

ciϕi and vh =∑

diϕi , then

Iterative solution

Corollary:

using the coercivity and boundedness of a,

|Ahc · d| ≤ M |c|Bh|d|Bh

, Ahc · c ≥ m |c|2Bhon RN .

That is,

|〈B−1h Ahc, d〉Bh

| ≤ M |c|Bh|d|Bh

, 〈B−1h Ahc, c〉Bh

≥ m |c|2Bh

⇒ Λ(B−1h Ah) ≤ M, λ(B−1

h Ah) ≥ m (mesh independent)

Iterative solution

Corollary:

That is,

| ≤ M |c|Bh|d|Bh

≥ m |c|2Bh

⇒ Λ(B−1h Ah) ≤ M, λ(B−1

Iterative solution

Corollary:

That is,

| ≤ M |c|Bh|d|Bh

≥ m |c|2Bh

⇒ Λ(B−1h Ah) ≤ M, λ(B−1

Iterative solution

Corollary: convergence of the PGCG-LS or PCGN iteration:

‖rk‖‖r0‖

1−(m

)2)k/2

or‖rk‖‖r0‖

≤ 2(M −m

)k⇒ mesh independent

⇒ optimal O(N) overall algorithm

Iterative solution

Corollary: convergence of the PGCG-LS or PCGN iteration:

‖rk‖‖r0‖

1−(m

)2)k/2

or‖rk‖‖r0‖

≤ 2(M −m

)k⇒ mesh independent

⇒ optimal O(N) overall algorithm

Convection-dominated problems

Finite element solution: convection-dominated case

The convection-dominated case: ε O(|w|)

→ standard FEM is problematic.

The convection-dominated case: ε O(|w|)→ standard FEM is problematic.

1. Boundary layers:

2. Deteriorating lower bound:

a(u, u) ≥ ε|u|21 (i.e. m = ε ≈ 0 ).

The convection-dominated case: ε O(|w|)→ standard FEM is problematic.

1. Boundary layers:

2. Deteriorating lower bound:

a(u, u) ≥ ε|u|21 (i.e. m = ε ≈ 0 ).

A widespread improvement: streamline diffusion FEM (SDFEM)

Choose paramaters δk > 0 on elements Tk ∈ TReplace test functions: vh → vh + δk w · ∇vh on Tk

⇒ stabilized bilinear form

aSD(uh, vh) :=

(ε∇uh·∇vh+(w·∇uh)vh

N∑k=1

(w · ∇uh) (w · ∇vh)

on Vh × Vh.

aSD(uh, vh) :=

N∑k=1

(w · ∇uh) (w · ∇vh)

on Vh × Vh.

aSD(uh, vh) :=

N∑k=1

(w · ∇uh) (w · ∇vh)

on Vh × Vh.

Stabilized problem: find uh ∈ Vh s.t.

aSD(uh, vh) = `SDvh (∀vh ∈ Vh).

Stabilized inner product: SD-inner product

〈uh, vh〉SD :=

∫Ωε∇uh · ∇vh +

N∑k=1

(w · ∇uh) (w · ∇vh) .

⇒ stable lower bound:

aSD(uh, uh) ≥ ‖uh‖2SD (i.e. m = 1 )!

Preconditioned CG iteration for the LAER:apply operator preconditioning!

Preconditioner = stiffness matrix for the SD-inner product:

(Sh)ij = 〈ϕi , ϕj〉SD

⇒ here 〈uh, vh〉SD =

(Lεuh) vh ,

where Lεu := −div(Aε∇u) with Aε = εI + δw ·wT

⇒ Sh corresponds to symmetric elliptic problems

⇒ optimal O(N) solvers available (multigrid, multilevel)

(Lεuh) vh ,

Convergence estimate → we need bounds m and M.

Seen above: m = 1.

Upper bound needed:

|aSD(uh, vh)| ≤ M ‖uh‖SD‖vh‖SD (∀uh, vh ∈ Vh)

‖uh‖2SD =

∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2 .

Seen above: m = 1.

Upper bound needed:

‖uh‖2SD =

∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2 .

Seen above: m = 1.

Upper bound needed:

‖uh‖2SD =

∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2 .

Seen above: m = 1.

Upper bound needed:

‖uh‖2SD =

∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2 .

Seen above: m = 1.

Upper bound needed:

‖uh‖2SD =

∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2 .

Standard Poincare-Friedrichs

Estimations

Estimation of the upper bound M.

aSD(uh, vh) :=

N∑k=1

(w · ∇uh) (w · ∇vh)︸︷︷︸〈uh,vh〉SD

(w·∇uh)vh

⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).

‖w · ∇uh‖2L2(Ω) ≤

N∑k=1

|w · ∇uh|2 ≤1

δ0‖uh‖2

Estimations

aSD(uh, vh) :=

N∑k=1

(w · ∇uh) (w · ∇vh)︸︷︷︸〈uh,vh〉SD

(w·∇uh)vh

⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).

‖w · ∇uh‖2L2(Ω) ≤

N∑k=1

|w · ∇uh|2 ≤1

δ0‖uh‖2

Estimations

aSD(uh, vh) :=

N∑k=1

(w · ∇uh) (w · ∇vh)︸︷︷︸〈uh,vh〉SD

(w·∇uh)vh

⇒ |aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)︸︷︷︸↓

‖vh‖L2(Ω).

‖w · ∇uh‖2L2(Ω) ≤

N∑k=1

|w · ∇uh|2 ≤1

δ0‖uh‖2

Estimations

Also needed: ‖vh‖L2(Ω) ≤ (?) · ‖vh‖SD . We follow [Kirby, 2010].

Squared:∫Ω|vh|2 ≤ (?)2 ·

(∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2).

Sufficient: ∫Ω|vh|2 ≤ (?)2 · ε

∫Ω|∇uh|2 .

Estimations

(∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2).

∫Ω|∇uh|2 .

Estimations

(∫Ωε |∇uh|2 +

N∑k=1

|w · ∇uh|2).

∫Ω|∇uh|2 .

Estimations

Squared:

∫Ω|vh|2 ≤ (?)2 ·

Ωε |∇vh|2 +

N∑k=1

|w · ∇vh|2︸︷︷︸≥0

∫Ω|∇vh|2 .

Estimations

Usual Poincare-Friedrichs inequality:∫Ω|vh|2 ≤ C 2

∫Ω|∇vh|2 .

⇒ ∫Ω|vh|2 ≤

ε· ε∫

Ω|∇vh|2 ≤

ε‖vh‖2

Estimations

Usual Poincare-Friedrichs inequality:∫Ω|vh|2 ≤ C 2

∫Ω|∇vh|2 .

⇒ ∫Ω|vh|2 ≤

ε· ε∫

Ω|∇vh|2 ≤

ε‖vh‖2

Estimations

Return to the estimate

|aSD(uh, vh)| ≤ ‖uh‖SD‖vh‖SD + ‖w · ∇uh‖L2(Ω)‖vh‖L2(Ω).

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

From Poincare-Friedrichs:

‖vh‖L2(Ω) ≤CΩ√ε‖vh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

1 +CΩ√δ0ε

)‖uh‖SD‖vh‖SD .

Estimations

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

1 +CΩ√δ0ε

Estimations

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

1 +CΩ√δ0ε

Estimations

Upper bound M:

|aSD(uh, vh)| ≤(

1 +CΩ√δ0ε

)︸︷︷︸

‖uh‖SD‖vh‖SD .

Problem:

1 +CΩ√δ0ε

)≈ ∞ if ε ≈ 0.

Estimations

Upper bound M:

|aSD(uh, vh)| ≤(

1 +CΩ√δ0ε

)︸︷︷︸

‖uh‖SD‖vh‖SD .

Problem:

1 +CΩ√δ0ε

)≈ ∞ if ε ≈ 0.

Streamline Poincare-Friedrichs

Estimations

Can it be improved?

New estimate: compare ‖uh‖L2(Ω) and ‖w · ∇uh‖L2(Ω)

→ streamline Poincare-Friedrichs inequality.

Estimations

Can it be improved?

New estimate: compare ‖uh‖L2(Ω) and ‖w · ∇uh‖L2(Ω)

→ streamline Poincare-Friedrichs inequality.

Assumption on the vector field w: global rectifiability(sufficient but not necessary for the results)

Definition. The vector field w ∈ C 1(Ω,Rn) is globally rectifiable onΩ if there exists a diffeomorphism h : K → Ω on a compact set Ksuch that

h(s1, . . . , sn−1, t) := γs1,...,sn−1(t)((s1, . . . , sn−1, t) ∈ K

)where t 7→ γs1,...,sn−1(t) are the family of characteristic curvescovering Ω.

Assumption on the vector field w: global rectifiability(sufficient but not necessary for the results)

Definition. The vector field w ∈ C 1(Ω,Rn) is globally rectifiable onΩ if there exists a diffeomorphism h : K → Ω on a compact set Ksuch that

h(s1, . . . , sn−1, t) := γs1,...,sn−1(t)((s1, . . . , sn−1, t) ∈ K

)where t 7→ γs1,...,sn−1(t) are the family of characteristic curvescovering Ω.

Figure : A globally rectifiable vector field.

Theorem. (Streamline Poincare-Friedrichs inequality). Letw ∈ C 1(Ω,Rn), for which w(x) 6= 0 (x ∈ Ω), be a globallyrectifiable vector field on Ω. Then there exists a constant Cw > 0(depending on w but independent of v) such that

‖v‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) (v ∈ H10 (Ω)).

Essence of proof:

Usual Poincare-Friedrichs:

‖v‖L2(Ω) ≤ c · ‖∂1v‖L2(Ω)︸︷︷︸Newton–Leibniz

≤ c · ‖∇v‖L2(Ω)

Streamline Poincare-Friedrichs:

‖v‖L2(Ω) ≤ Cw · ‖∂wv‖L2(Ω)︸︷︷︸local change of variables + streamline Newton–Leibniz

= Cw · ‖w · ∇v‖L2(Ω)

Theoretical estimates

Robust estimations

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

From streamline Poincare-Friedrichs:

‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw

δ0‖vh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

Robust estimations

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw

δ0‖vh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

Robust estimations

We have seen

‖w · ∇uh‖L2(Ω) ≤1√δ0‖uh‖SD .

‖vh‖L2(Ω) ≤ Cw ‖w · ∇v‖L2(Ω) ≤Cw

δ0‖vh‖SD .

Altogether:

|aSD(uh, vh)| ≤(

Robust estimations

That is: the upper bound of aSD satisfies

M ≤ 1 +Cw

independently of ε.

Consequence: the PCG iterations converge with rate independentlyof ε → robustness.

Robust estimations

That is: the upper bound of aSD satisfies

M ≤ 1 +Cw

independently of ε.

Consequence: the PCG iterations converge with rate independentlyof ε → robustness.

Estimates and experiments

Numerical experiments

u|∂Ω = 0.

domain Ω := [0, 1]2 in R2.

w := (1, 0) a fixed constant vector

exact solution u(x , y) =(

x − ex/ε−1e1/ε−1

)4y(1− y)

→ boundary layer near the segment x = 1.

Figure : result for ε = 10−10 – no unphysical oscillations.J. Karatson Department of Applied Analysis, ELTE

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