Nitsche-Mortaring for Singularly Perturbed Convection-Diffusion - - PowerPoint PPT Presentation

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Nitsche-Mortaring for Singularly Perturbed Convection-Diffusion - - PowerPoint PPT Presentation

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Background Theoretical results Numerical experiments Nitsche-Mortaring for Singularly Perturbed Convection-Diffusion Problems Martin Schopf Joint work with Torsten Lin and Hans-G org Roos Fachrichtung Mathematik Institut f ur

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SLIDE 1

Background Theoretical results Numerical experiments

Nitsche-Mortaring for Singularly Perturbed Convection-Diffusion Problems

Martin Schopf

Joint work with Torsten Linß and Hans-G¨

  • rg Roos

Fachrichtung Mathematik Institut f¨ ur Numerische Mathematik

  • 1. Mai 2010

1 / 26

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SLIDE 2

Background Theoretical results Numerical experiments

Convection-diffusion problem

Consider:

✁ε∆u ✁ b ☎ ∇u cu ✏ f

in

Ω,

u ✏ 0

  • n

❇Ω,

(P) where b, c, f are smooth, 0 ➔ ε ✦ 1 and b → ♣β1, β2q → 0, c 1 2∇ ☎ b ➙ c0 → 0. Outflow boundary: exponential layers of width O♣ε ln 1④εq

2 / 26

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SLIDE 3

Background Theoretical results Numerical experiments

Singular perturbation and layers

Solution for ε ✏ 10✁3

3 / 26

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Background Theoretical results Numerical experiments

Singular perturbation and layers

Oscillations in the result of standard methods (ε ✏ 10✁3)

4 / 26

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SLIDE 5

Background Theoretical results Numerical experiments

Layer adapted discretization

Shishkin-mesh, bilinear Galerkin-FEM Error of bilinear Galerkin-FEM an a 64 ✂ 64 Shishkin-mesh

5 / 26

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SLIDE 6

Background Theoretical results Numerical experiments

Layer adapted discretization

Shishkin-mesh, linear Galerkin-FEM Error of linear Galerkin-FEM

  • n a 64 ✂ 64 Shishkin-mesh

6 / 26

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SLIDE 7

Background Theoretical results Numerical experiments

Nitsche mortaring

Domain decomposition: ¯

Ω ✏ ¯ Ω1 ❨ ¯ Ω2

Interface: Γ ✏ ¯

Ω1 ❳ ¯ Ω2

Ω1 Ω2 Γ n1 n2

Interface problem:

✁ε∆ui ✁ b ☎ ∇ui cui ✏ fi

in

Ωi, i ✏ 1, 2,

ui ✏ 0

  • n

❇Ω ❳ ❇Ωi, i ✏ 1, 2,

u1 ✏ u2

  • n

Γ, ❇u1 ❇n1 ❇u2 ❇n2 ✏ 0

  • n

Γ,

(IP)

7 / 26

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SLIDE 8

Background Theoretical results Numerical experiments

Notation

  • Restriction: vi ≔ v⑤Ωi
  • Broken Sobolev space:

Ω1 Ω2 Γ n1 n2

V ≔ tv P L 2♣Ωq : vi P H1♣Ωiq ❅i P t1, 2✉, v⑤❇Ω ✏ 0✉

  • Jump along the interface: ✈v✇ ≔ v1⑤Γ ✁ v2⑤Γ
  • Assumption: n2 ☎ b ↕ 0
  • n Γ

8 / 26

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SLIDE 9

Background Theoretical results Numerical experiments

Weak formulation

Find u P V, such that aσ

M♣u, vq ✏ ♣f, vqΩ

❅v P V,

where aσ

M♣u, vq ≔ aG♣u, vq aσ I ♣u, vq,

aG♣u, vq ≔

2

➳

i✏1

ε ♣∇ui, ∇viqΩi ✁

2

➳

i✏1

♣b ☎ ∇ui ✁ cui, viqΩi ,

aσ

I ♣u, vq ≔ ✁ε

  • α1

❇u1 ❇n1 ✁ α2 ❇u2 ❇n2 , ✈v✇

  • Γ

σ ε

  • α1

❇v1 ❇n1 ✁ α2 ❇v2 ❇n2 , ✈u✇

  • Γ

✁ ♣b ☎ n2✈u✇, v1qΓ ♣γN✈u✇, ✈v✇qΓ.

9 / 26

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SLIDE 10

Background Theoretical results Numerical experiments

Weak formulation

  • ☎,☎Γ denotes duality pairing on H✁1④2

00

♣Γq ✂ H

1④2 00 ♣Γq

  • αi ➙ 0, constants with α1 α2 ✏ 1
  • σ ✏

✩ ✬ ✫ ✬ ✪ ✁1

symmetric mortaring incomplete 1 nonsymmetric

  • γN, positive constant (only depends on N)

aσ

I ♣u, vq ≔ ✁ε

  • α1

❇u1 ❇n1 ✁ α2 ❇u2 ❇n2 , ✈v✇

  • Γ

σ ε

  • α1

❇v1 ❇n1 ✁ α2 ❇v2 ❇n2 , ✈u✇

  • Γ

✁ ♣b ☎ n2✈u✇, v1qΓ ♣γN✈u✇, ✈v✇qΓ.

10 / 26

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SLIDE 11

Background Theoretical results Numerical experiments

Model problem

  • Problem (P) on the unit square, Ω ✏ ♣0, 1q2
  • Shishkin type mesh with transition point λ ✏ ♣λ1, λ1q and

λi ✏ 2ε

βi ln N

  • N, number of mesh intervals in each coordinate direction

n Ω2 Ω1 Γ λ

2

  • 11 / 26
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Background Theoretical results Numerical experiments

Discretization

Find uh P Vh⑨V , such that aσ

M♣uh, vhq ✏ ♣f, vhqΩ

❅vh P Vh

where Vh ≔

✥

v P V : v⑤T P Q1♣Tq

❅T P T ♣Ω1q,

v⑤T P P1♣Tq

❅T P T ♣Ω2q ✭ .

h1 H1 h2 H2 ¯ h

N✁1 ➔ H1, H2 ➔ 2N✁1 “long” edge h1, h2 ✏ O♣εN✁1 ln Nq “short” edge C1N✁1➔ ➔C2N✁1

max diameter

  • f a triangle

12 / 26

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SLIDE 13

Background Theoretical results Numerical experiments

Properties of the discretization

Galerkin orthogonality

aσ

M♣u ✁ uh, vhq ✏ 0

❅vh P Vh

Coercivity, nonsymmetric mortaring

We have a1

M ♣v, vq ➙ ⑦v⑦2

❅v P V,

with the broken energy norm:

⑦v⑦2 ≔ ε

2

➳

i✏1

⑤vi⑤2

1,Ωic0 2

➳

i✏1

⑥vi⑥2

0,Ωi♣γN ✈v✇,✈v✇qΓ✁1

2 ♣b ☎ n2 ✈v✇, ✈v✇qΓ

13 / 26

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SLIDE 14

Background Theoretical results Numerical experiments

Properties of the discretization

Coercivity in Vh

Suppose

γN ➙ ˜

C

★ ε✁1

if α1 ✏ 0 and N♣ln Nq✁1 if α1 P ♣0, 1s. Then there exists a constant C → 0 such that aσ

M♣vh, vhq ➙ C⑦vh⑦2

❅vh P Vh.

14 / 26

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SLIDE 15

Background Theoretical results Numerical experiments

Solution decomposition

Assume that the solution u of (P) can be decomposed: u ✏ S E1 E2 E12, (SD)

✞ ✞ ✞ ✞ ❇ij ❇xi❇yj S♣x, yq ✞ ✞ ✞ ✞ ↕ C ✞ ✞ ✞ ✞ ❇ij ❇xi❇yj E1♣x, yq ✞ ✞ ✞ ✞ ↕ Cε✁ie✁β1x④ε ✞ ✞ ✞ ✞ ❇ij ❇xi❇yj E2♣x, yq ✞ ✞ ✞ ✞ ↕ Cε✁je✁β2y④ε ✞ ✞ ✞ ✞ ❇ij ❇xi❇yj E12♣x, yq ✞ ✞ ✞ ✞ ↕ Cε✁♣ijqe✁♣β1xβ2yq④ε

for ♣x, yq P ¯

Ω,

0 ↕ i j ↕ 3

15 / 26

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SLIDE 16

Background Theoretical results Numerical experiments

Interpolation error estimates

Suppose ε1④2 ↕ ♣ln Nq✁2 and (SD). Then the interpolation error on our Shishkin mesh satisfies

✎ ✎u ✁ uI✎ ✎

✽,Ω1 ↕ C♣N✁2 ln2 Nq,

✎ ✎u ✁ uI✎ ✎

✽,Ω2 ↕ CN✁2,

ε ✎ ✎∇

  • u ✁ uI✟✎

✎

✽,Ω1 ↕ C♣N✁1 ln Nq,

ε ✎ ✎∇

  • u ✁ uI✟✎

✎

✽,Ω2 ↕ CN✁1,

✎ ✎u ✁ uI✎ ✎

0,Ω1

✎ ✎u ✁ uI✎ ✎

0,Ω2 ↕ CN✁2,

ε1④2 ✎ ✎∇

  • u ✁ uI✟✎

✎

0,Ω1 ε1④2 ✎

✎∇

  • u ✁ uI✟✎

✎

0,Ω2 ↕ CN✁1 ln N,

✎ ✎✈u ✁ uI✇ ✎ ✎

0,Γ ↕ 2

✎ ✎✈u ✁ uI✇ ✎ ✎

✽,Γ ↕ C♣N✁2 ln2 Nq.

16 / 26

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SLIDE 17

Background Theoretical results Numerical experiments

Error analysis

Supercloseness theorem

Assumptions:

  • Solution decomposition (SD),
  • Triangulation T ♣Ω2q shape regular with parameter ,

satisfying ✒ N✁1

  • Mortaring parameter γN ✏ γN④ ln N, γ sufficiently large,
  • ⑤aG♣η, ξq⑤ ↕ CN✁3④2 ln2 N⑦ξ⑦,

then

⑦u ✁ uh⑦ ↕ ⑦ u ✁ uI ❧♦ ♦♠♦ ♦♥

η

⑦ ⑦ uI ✁ uh ❧♦♦♠♦♦♥

ξ

⑦ ↕ C

  • N✁1 ln N N✁3④2 ln2 N

✟ .

17 / 26

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SLIDE 18

Background Theoretical results Numerical experiments

Proof (outline)

C⑦ξ⑦2 ↕ aσ

M♣η, ξq ✏ aG♣η, ξq aσ I ♣η, ξq

✏

2

➳

i✏1

ε ♣∇ηi, ∇ξiqΩi ✁

2

➳

i✏1

♣b ☎ ∇ηi ✁ cηi, ξiqΩi ✁ ε ✂ α1 ❇η1 ❇n1 ✁ α2 ❇η2 ❇n2 , ✈ξ✇ ✡

Γ

(I1)

σ ε ✂ α1 ❇ξ1 ❇n1 ✁ α2 ❇ξ2 ❇n2 , ✈η✇ ✡

Γ

(I2)

✁ ♣b ☎ n2 ✈η✇,ξ1qΓ

(I3)

♣γN ✈η✇, ✈ξ✇qΓ

(I4)

18 / 26

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SLIDE 19

Background Theoretical results Numerical experiments

Proof (outline)

⑦ξ⑦ ↕ C ✂⑤aG♣η, ξq⑤ ⑦ξ⑦

Galerkin part,

γ✁1④2

N

  • α1N✁1 ln N α2N✁1✟

from (I1)

N✁3④2 α1 ln3④2 N α2ε1④2 ln2 N ✟

from (I2)

N✁2 ln2 N ✁ γ✁1④2

N

N1④2✠

from (I3)

γ

1④2 N N✁2 ln2 N

✡

from (I4)

↕ CN✁3④2 ln2 N

for

γN ✏ γN④ ln N

19 / 26

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SLIDE 20

Background Theoretical results Numerical experiments

Error analysis for different asymptotic behavior

Supercloseness theorem

Assumptions:

  • Solution decomposition (SD),
  • Triangulation T ♣Ω2q shape regular with parameter ,

satisfying ✒ N✁1ln N

  • Mortaring parameter γN ✏ γN④ ln N, γ sufficiently large,
  • ⑤aG♣η, ξq⑤ ↕ CN✁3④2 ln 3④2N⑦ξ⑦,

then

⑦u ✁ uh⑦ ↕ ⑦ u ✁ uI ❧♦ ♦♠♦ ♦♥

η

⑦ ⑦ uI ✁ uh ❧♦♦♠♦♦♥

ξ

⑦ ↕ C

  • N✁1 ln N N✁3④2 ln 3④2N

✟ .

20 / 26

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SLIDE 21

Background Theoretical results Numerical experiments

Error analysis for different asymptotic behavior

Convergence theorem

Assumptions:

  • Solution decomposition (SD),
  • Triangulation T ♣Ω2q shape regular with parameter ,

satisfying ✒ N✁3④4 ln3④4 N

  • Mortaring parameter γN ✒ N ln ✁3④4N,
  • ⑤aG♣η, ξq⑤ ↕ CN✁1 ln N⑦ξ⑦,

then

⑦u ✁ uh⑦ ↕ CN✁1 ln N.

21 / 26

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SLIDE 22

Background Theoretical results Numerical experiments

Test problem

  • singularly perturbed problem:

✁ε ∆u ✁ 2ux ✁ 3uy u ✏ f in Ω ✏ ♣0, 1q2

u ✏ 0 on ❇Ω.

  • exact solution:

u♣x, yq ✏ 2 sin♣1 ✁ xq

  • 1 ✁ e✁2x④ε✟

♣1 ✁ yq2

1 ✁ e✁3y④ε✟

22 / 26

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SLIDE 23

Background Theoretical results Numerical experiments

FEM Solution

  • hybrid tensor product

mesh of regular type

(N ✏ 8)

error on this mesh

(N ✏ 64, ε ✏ 10✁8, σ ✏ ✁1, α1 ✏ 0,

γN ✏ 0.5N④ ln N)

23 / 26

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SLIDE 24

Background Theoretical results Numerical experiments

Uniformity: error and ⑦ξ⑦

N ε ✏ 10✁4 ε ✏ 10✁8 ε ✏ 10✁12 error rate error rate error rate 8 4.3357e-1 0.5673 0.6652 0.7314 0.7755 0.8066 4.3357e-1 0.5673 0.6653 0.7314 0.7756 0.8066 4.3357e-1 0.5673 0.6653 0.7314 0.7756 0.8066 16 2.9260e-1 2.9260e-1 2.9260e-1 32 1.8450e-1 1.8450e-1 1.8450e-1 64 1.1113e-1 1.1113e-1 1.1113e-1 128 6.4914e-2 6.4914e-2 6.4914e-2 256 3.7111e-2 3.7113e-2 3.7113e-2 N ε ✏ 10✁4 ε ✏ 10✁8 ε ✏ 10✁12 ⑦ξ⑦ rate ⑦ξ⑦ rate ⑦ξ⑦ rate 8 8.9255e-2 1.2661 1.3099 1.4178 1.5160 1.5972 8.9222e-2 1.2671 1.3128 1.4236 1.5239 1.5973 8.9222e-2 1.2671 1.3128 1.4236 1.5239 1.5973 16 3.7107e-2 3.7072e-2 3.7072e-2 32 1.4966e-2 1.4923e-2 1.4923e-2 64 5.6008e-3 5.5630e-3 5.5630e-3 128 1.9582e-3 1.9345e-3 1.9345e-3 256 6.4716e-4 6.3934e-4 6.3933e-4

error and ⑦ξ⑦ in broken energy norm (σ ✏ ✁1, α1 ✏ 1④2 and γN ✏ 0.5N ln✁1 N)

24 / 26

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SLIDE 25

Background Theoretical results Numerical experiments

Chevron grid

  • hybrid tensor product

mesh of chevron type (N ✏ 8) error on such a mesh

(N ✏ 64, ε ✏ 10✁8, σ ✏ 1, α1 ✏ 0.5,

γN ✏ 0.5N④ ln N)

25 / 26

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SLIDE 26

Background Theoretical results Numerical experiments

Sharp error estimate

N

⑦u ✁ uh⑦

rate

⑦u ✁ uh⑦ N

ln N

⑦ξ⑦

rate

⑦ξ⑦ N3④2

ln3④4 N

8 5.7001e-1 0.5523 0.6599 0.7282 0.7749 0.8066 2.1929 1.9464e-1 1.0960 1.2995 1.3096 1.3545 1.2919 2.5434 16 3.8872e-1 2.2432 9.1054e-2 2.7122 32 2.4603e-1 2.2717 3.6992e-2 2.6363 64 1.4852e-1 2.2855 1.4924e-2 2.6238 128 8.6796e-2 2.2897 5.8362e-3 2.5852 256 4.9622e-2 2.2909 2.3836e-3 2.7018 error and ⑦ξ⑦ in broken energy norm (σ ✏ ✁1, α1 ✏ 1④2 and γN ✏ 0.5N ln✁1 N

26 / 26

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SLIDE 27

Different asymptotic behavior (1)

N ⑦u ✁ uh⑦ rate ⑦u ✁ uh⑦ N

ln N

⑦ξ⑦ rate ⑦ξ⑦ N2

ln2 N

8 9.7102e-001 0.5383 0.6542 0.7270 0.7737 0.8060 3.7357 3.0897e-001 1.1175 1.3274 1.4848 1.5452 1.6076 4.5730 16 6.6861e-001 3.8584 1.4240e-001 4.7423 32 4.2484e-001 3.9227 5.6746e-002 4.8378 64 2.5668e-001 3.9500 2.0275e-002 4.8013 128 1.5014e-001 3.9608 6.9471e-003 4.8348 256 8.5875e-002 3.9645 2.2796e-003 4.8586

error and ⑦ξ⑦ ✏ ⑦uI ✁ uh⑦ in broken energy norm non-matching grid sequence of regular type symmetric mortaring (σ ✏ ✁1, α1 ✏ 1④2, γN ✏ 0.5N ln✁1 N and ✒ N✁1 ln N)

27 / 26

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SLIDE 28

Different asymptotic behavior (2)

N ⑦u ✁ uh⑦ rate ⑦u ✁ uh⑦ N

ln N

⑦ξ⑦ rate ⑦ξ⑦ N2

ln2 N

8 9.7026e-001 0.5380 0.6548 0.7260 0.7734 0.8059 3.7328 2.9401e-001 1.1387 1.3546 1.4585 1.5471 1.5845 4.3517 16 6.6824e-001 3.8563 1.3353e-001 4.4469 32 4.2444e-001 3.9189 5.2219e-002 4.4518 64 2.5661e-001 3.9489 1.9001e-002 4.4997 128 1.5012e-001 3.9603 6.5019e-003 4.5249 256 8.5872e-002 3.9644 2.1679e-003 4.6206

error and ⑦ξ⑦ ✏ ⑦uI ✁ uh⑦ in broken energy norm non-matching grid sequence of regular type symmetric mortaring (σ ✏ ✁1, α1 ✏ 1④2, γN ✏ 0.5N ln✁3④4 N and ✒ N✁3④4 ln3④4 N)

28 / 26

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SLIDE 29

Irregular meshes

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Delaunay triangulation N ✏ 16 N

⑦u ✁ uh⑦

rate

⑦ξ⑦

rate 16 3.2526e-1 0.7944 0.7462 0.7820 0.8078 1.9022e-1 1.5061 1.6902 1.4173 1.6429 32 1.8754e-1 6.6966e-2 64 1.1181e-1 2.0752e-2 128 6.5023e-2 7.7699e-3 256 3.7146e-2 2.4879e-3 error and convergence rates in broken energy norm (σ ✏ 1, α1 ✏ 1④2 and γN ✏ 0.5N④ ln N)

29 / 26

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SLIDE 30

Comparison in broken energy norm

N

  • nonsym. mortaring
  • sym. mortaring

⑦u ✁ uh⑦ ⑦ξ⑦ ⑦u ✁ uh⑦ ⑦ξ⑦

8 4.5776e-1 1.6173e-1 4.3357e-1 8.9222e-2 16 2.9299e-1 3.9338e-2 2.9260e-1 3.7072e-2 32 1.8447e-1 1.4533e-2 1.8450e-1 1.4923e-2 64 1.1112e-1 5.4854e-3 1.1113e-1 5.5630e-3 128 6.4914e-2 1.9232e-3 6.4914e-2 1.9345e-3 256 3.7113e-2 6.3778e-4 3.7113e-2 6.3934e-4 error in broken energy norm (α1 ✏ 1④2 and γN ✏ 0.5N④ ln N)

30 / 26

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SLIDE 31

Comparison in broken energy norm

N

α1 ✏ 0 α1 ✏ 0.5 ⑦u ✁ uh⑦ ⑦ξ⑦ ⑦u ✁ uh⑦ ⑦ξ⑦

8 4.3462e-1 8.7071e-2 4.3357e-1 8.9222e-2 16 2.9247e-1 3.5963e-2 2.9260e-1 3.7072e-2 32 1.8449e-1 1.4736e-2 1.8450e-1 1.4923e-2 64 1.1112e-1 5.5349e-3 1.1113e-1 5.5630e-3 128 6.4914e-2 1.9300e-3 6.4914e-2 1.9345e-3 256 3.7113e-2 6.3865e-4 3.7113e-2 6.3934e-4 error in broken energy norm with sym. mortarting (σ ✏ ✁1 and γN ✏ 0.5N④ ln N)

31 / 26

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SLIDE 32

Interpolation error estimates (general case)

Suppose ε1④2 ↕ ♣ln Nq✁2. The interpolation error on our Shishkin mesh satisfies

✎ ✎u ✁ uI✎ ✎

✽,Ω1 ↕ C♣N✁2 ln2 Nq,

✎ ✎u ✁ uI✎ ✎

✽,Ω2 ↕ C♣N✁2 2q,

ε ✎ ✎∇

  • u ✁ uI✟✎

✎

✽,Ω1 ↕ C♣N✁1 ln Nq,

ε ✎ ✎∇

  • u ✁ uI✟✎

✎

✽,Ω2 ↕ C♣N✁1 N✁2♣ln Nq✁4✁1 ♣ln Nq✁4q,

✎ ✎u ✁ uI✎ ✎

0,Ω1 ↕ CN✁2,

✎ ✎u ✁ uI✎ ✎

0,Ω2 ↕ C2,

ε1④2 ✎ ✎∇

  • u ✁ uI✟✎

✎

0,Ω1 ↕ CN✁1 ln N,

ε1④2 ✎ ✎∇

  • u ✁ uI✟✎

✎

0,Ω2 ↕ C♣N✁2 ♣ln Nq✁2 N✁2♣ln Nq✁2✁1q,

✎ ✎✈u ✁ uI✇ ✎ ✎

0,Γ ↕ 2

✎ ✎✈u ✁ uI✇ ✎ ✎

✽,Γ ↕ C♣N✁2 ln2 N 2q.

32 / 26

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SLIDE 33

Estimate for ξ (general case)

⑦ξ⑦ ↕ C ✂⑤aG♣η, ξq⑤ ⑦ξ⑦

Galerkin,

α1γ✁1④2

N

N✁1 ln N

α2γ✁1④2

N

✁

N✁1 N✁2 ♣ln Nq✁4 ✁1 ♣ln Nq✁4

✠

from (I1),

α1 ✁

N✁3④2 ln3④2 N N1④2 ♣ln Nq✁1④2 2✠

ε1④2N✁2 ln2 N✁1④2 ε1④23④2

from (I2),

  • N✁2 ln2 N 2✟ ✁

γ✁1④2

N

✁1④2✠

from (I3),

γ

1④2 N

  • N✁2 ln2 N 2✟ ✡

from (I4).

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slide-34
SLIDE 34

Important tool

Trace lemma in finite dimensional spaces

Let T triangle (rectangle), F ⑨ T edge of T and h❑

F height of

the element T above the side F. For a function p P Pt♣Tq (p P Qt♣Tq) with t P N the estimate

p2

0,F ↕ C 1

h❑

F

p2

0,T

  • holds. Here C depends only on the polynomial degree t.

proof uses:

  • Trace lemma on reference element
  • Inverse estimate
  • Transformation

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