Three remarks on anisotropic finite elements Thomas Apel Universit - - PowerPoint PPT Presentation

Three remarks on anisotropic finite elements

Thomas Apel

Universit¨ at der Bundeswehr M¨ unchen

Workshop Numerical Analysis for Singularly Perturbed Problems dedicated to the 60th birthday of Martin Stynes

Apel 1 / 34

Instead of a motivation of anisotropic finite elements

Congratulations to Martin!

Apel 2 / 34

Plan of the talk

1

A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch)

2

Remarks on interpolation

Apel 3 / 34

Plan of the talk

1

A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch)

2

Remarks on interpolation

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 4 / 34

Classical formulation

Lu := −div (A∇u) + b · ∇u + cu = f in Ω u =

• n ΓD

A∇u · n = g

• n ΓN

Layers due to dominating convection b:

• rdinary boundary layers at outflow boundary

parabolic boundary layers due to flow parallel to the boundary Layers due to discontinuous right hand side f: internal layer due to small diffusion through line/face of discontinuity Layers due to anisotropic diffusion tensor A: e.g. A = ε 1

• : diffusion small in x-direction: layers left and right

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 5 / 34

Example for illustrating anisotropic diffusion

−div (A∇u) = f in Ω = (0, 1)2 u =

• n Γ

f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Apel

A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 6 / 34

Example: α = 0

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = ε 1

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 7 / 34

Example: α = 0.5π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = 1 ε

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 8 / 34

Example: α = 0.49π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 9 / 34

Example: α = 0.48π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 10 / 34

Example: α = 0.47π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 11 / 34

Example: α = 0.46π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 12 / 34

Example: α = 0.45π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A = cos α − sin α sin α cos α ε 1 cos α sin α − sin α cos α

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 13 / 34

Example: α = 0.25π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A =

1 2

• 1 + ε

1 − ε 1 − ε 1 + ε

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 14 / 34

Example: α = 0.05π

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A =

• ε

1

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 15 / 34

Example: α = 0

−div (A∇u) = f in Ω = (0, 1)2 f =

• −1

for 0 < x < 1

2

+1 for 1

2 < x < 1

A =

• ε

1

• Picture by G. Winkler

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 16 / 34

Related results

a priori error analysis (anisotropic diffusion): Li, Wheeler a posteriori error estimation (convection dominated problems): isotropic: Angermann, Kay/Silvester, Sangalli, Verf¨ urth anisotropic: Formaggia/Perotto/Zunino, Kunert, Picasso a posteriori error estimation (anisotropic diffusion): Fierro/Veeser

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 17 / 34

SUPG discretization

Let Vh := {v ∈ C(Ω) : v|T ∈ P1(T) ∀T ∈ Th, v = 0 on ΓD}, Bh(uh, vh) = (A∇uh · ∇vh) + (b · ∇uh + cuh, vh) +

• T

δT(Luh, b · ∇vh)T Fh(vh) = (f, vh) + (g, vh)ΓN +

• T

δT(f, b · ∇vh)T The parameters δT ≥ 0 satisfy δT ≤ min{µ−2h2

min,TA1/2−1 2→2, c0c2−1 ∞,T, hmin,A,TA−1/2b−1 ∞,T}

where µ is the constant in some inverse inequality and hmin,A,T = min

T ′⊂ωT hmin,FA(T ′) with FA(x) = A−1/2x.

Streamline upwind Petrov-Galerkin (SUPG) scheme: Find uh ∈ Vh with Bh(uh, vh) = Fh(vh) ∀vh ∈ Vh.

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 18 / 34

Error estimation

Residuals: element residual: RT := f − Luh edge residual: RE := [A∇uh · nE] with modification on ∂Ω Resulting error estimator: η2

T := α2 TrT2 T +

• E∈∂T\ΓD

αEβ−1

E rE2 E,

η2 :=

• T

η2

T

where αT = min{c−1/2 , hmin,A,T}, βE = maxT⊂ωE(hE,Th−1

min,A,T),

αE = αT corresponding to βE. Approximation terms: ζ2

T := α2 T

• T⊂ωT

RT ′ − rT ′2

T ′ +

• E∈∂T\ΓD

αEβ−1

E RE − rE2 E,

ζ2 :=

• K∈Th

ζ2

T.

Weighted norm: ||| u |||2 := (A∇u, ∇u)2 + (c0u, u) Dual norm: ||| φ |||∗ := sup

v∈H1

ΓD (Ω)\{0}

• Ω φv

||| v ||| =:

• Ω

φv1.

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 19 / 34

Quality of the error estimator

Theorem: [Apel/Nicaise/Sirch:11] The error is bounded from above by ||| u − uh ||| m1(u − uh, A, Th) · (η + ζ) ||| b · ∇(u − uh) |||∗ κ||| u − uh ||| + m1(v1, A, Th)(η + ζ) and from below by η κ||| u − uh ||| + ||| b · ∇(u − uh) |||∗ + ζ where κ = max{1, c−1

0 c∞,T ′}.

Alignment measure: m2

1(v, A, Th) :=

• T

h−2

min,A,TC⊤ A,T ∇v2 T

A1/2∇v2 Robust lower bound was obtained due to the use of the dual norm of the convective derivative, cf. also [Verf¨ urth 05] for isotropic meshes.

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 20 / 34

Note on alignment

Our a posteriori error estimates are reliable if some alignment measure is close to unity: no problem if the aspect ratio is smaller than appropriate too much anisotropy increases the value of the alignment measure If is optimal, then the following situations lead to an increased alignment measure:

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 21 / 34

Numerical test: the problem

−div (A∇u) + b · ∇u = f in Ω = (0, 1)2 u = g

• n Γ

with A =

• ε

1

• ,

b =

• 1
• Data f and g are chosen such that

u = 10y(1 − y)

• e−x − e−1+(x−1)/ε

, is the exact solution. The solution contains a typical boundary layer of that problem. Both the data and the solution are O(1) in the L2(Ω)- and L∞(Ω)-norms uniformly in ε.

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 22 / 34

Numerical test: the discretization

Mesh: piecewise uniform with an anisotropic part in the boundary strip ΩL = (1 − 2ε| ln ε|, 1) × (0, 1). SUPG scheme with δT = √εh2

T,min

in the boundary layer, h2

T,min

elsewhere. is chosen.

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 23 / 34

Numerical test: behaviour of the error

. . . in the max norm and in the “SUPG norm” |[v]|2 = B(v, v) +

T∈Th δTb · ∇v2 L2(T).

ε = 10−4 N eL∞(Ω) rate |[e]| rate 153 1.62 E−01 6.86 E−01 561 7.12 E−02 1.27 3.16 E−01 1.19 2145 2.57 E−02 1.52 1.53 E−01 1.08 8385 7.89 E−03 1.73 7.61 E−02 1.03 33153 2.20 E−03 1.86 3.80 E−02 1.01 131841 5.81 E−04 1.93 1.90 E−02 1.00 525825 1.49 E−04 1.96 9.48 E−03 1.00 ε = 10−8 N eL∞(Ω) rate |[e]| rate 153 3.42 E−01 5.35 E+00 561 1.64 E−01 1.14 1.78 E+00 1.69 2145 6.92 E−02 1.29 5.19 E−01 1.84 8385 2.50 E−02 1.49 1.60 E−01 1.73 33153 7.68 E−03 1.72 5.97 E−02 1.43 131841 2.14 E−03 1.85 2.67 E−02 1.17 525825 5.65 E−04 1.92 1.29 E−02 1.05

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 24 / 34

Numerical test: behaviour of the error estimator

ε = 10−4 N η rate ||| e ||| rate ||| b · ∇e |||∗ rate Ieff 153 3.10 E+00 5.86 E−01 7.73 E−02 4.68 561 1.63 E+00 0.988 3.00 E−01 1.03 2.44 E−02 1.77 5.03 2145 8.46 E−01 0.981 1.51 E−01 1.02 6.85 E−03 1.89 5.35 8385 4.33 E−01 0.983 7.58 E−02 1.01 1.86 E−03 1.91 5.58 33153 2.20 E−01 0.988 3.79 E−02 1.01 4.95 E−04 1.93 5.72 131841 1.11 E−01 0.993 1.90 E−02 1.00 1.28 E−04 1.96 5.79 525825 5.55 E−02 0.996 9.48 E−03 1.00 3.23 E−05 1.99 5.84 ε = 10−8 N η rate ||| e ||| rate ||| b · ∇e |||∗ rate Ieff 153 3.69 E+00 6.65 E−01 2.19 E−01 4.17 561 1.96 E+00 0.973 3.78 E−01 0.869 8.50 E−02 1.46 4.23 2145 1.04 E+00 0.939 2.00 E−01 0.952 2.52 E−02 1.81 4.64 8385 5.52 E−01 0.935 1.02 E−01 0.993 6.71 E−03 1.94 5.10 33153 2.87 E−01 0.950 5.10 E−02 1.00 1.74 E−03 1.97 5.45 131841 1.47 E−01 0.969 2.55 E−02 1.00 4.46 E−04 1.97 5.67 525825 7.46 E−02 0.982 1.28 E−02 1.00 1.14 E−04 1.97 5.80

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 25 / 34

Numerical test: computation of the dual norm

The error in the dual norm ||| φ |||∗ is approximately computed here by ||| φ |||∗ = sup

v∈H1

ΓD (Ω)\{0}

• Ω φv

||| v ||| ≈ sup

vh∈Vh\{0}

• Ω φhvh

||| vh ||| where φh is an approximation of φ in a finite dimensional space Wh, here the space of piecewise constants. By some linear algebra one can show that the latter expression is computable: sup

vh∈Vh\{0}

• Ω φhvh

||| vh ||| =

• φTMK −1MTφ

1/2 .

Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 26 / 34

Plan of the talk

1

A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch)

2

Remarks on interpolation

Apel Remarks on interpolation 27 / 34

Isotropic (shape-regular) and anisotropic elements

Isotropic elements: The aspect ratio enters the constant in certain estimates. Anisotropic elements: Estimates should be valid uniformly in the aspect ratio.

Apel Remarks on interpolation 28 / 34

Remarks on wrong conjectures I

Let T be an isotropic triangle, and Ih the piecewise linear interpolant w.r.t. the

• vertices. Then

|Ihu|H1(T) |u|H1(T) ∀u ∈ H1(T) ∩ C(T), u − IhuL2(T) hT|u|H1(T) ∀u ∈ H1(T) ∩ C(T).

Apel Remarks on interpolation 29 / 34

Remarks on wrong conjectures I

Let T be an isotropic triangle, and Ih the piecewise linear interpolant w.r.t. the

• vertices. Then

|Ihu|H1(T) |u|H1(T) ∀u ∈ H1(T) ∩ C(T), u − IhuL2(T) hT|u|H1(T) ∀u ∈ H1(T) ∩ C(T). Counterexample: Let T be the reference triangle and r =

• x2

1 + x2 2.

0.2 0.4 0.6 0.8 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y u

ε = 0.4

Picture by Th. Flaig

uε(x1, x2) = min

• 1, ε ln
• ln r

e

• Ihuε(x1, x2) = 1 − x1 − x2

independent of ε |uε|H1(T) → 0 for ε → 0 |Ihuε|H1(T) → 0 for ε → 0 |uε − Ihuε|L2(T) → 0 for ε → 0 Origin of this example?

Apel Remarks on interpolation 29 / 34

Remarks on wrong conjectures I

Let T be an isotropic triangle, and Ih the piecewise linear interpolant w.r.t. the

• vertices. Then

|Ihu|H1(T) |u|H1(T) ∀u ∈ H1(T) ∩ C(T), u − IhuL2(T) hT|u|H1(T) ∀u ∈ H1(T) ∩ C(T). Counterexample: Let T be the reference triangle and r =

• x2

1 + x2 2.

0.2 0.4 0.6 0.8 1 0.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x y u

ε = 0.4

Picture by Th. Flaig

uε(x1, x2) = min

• 1, ε ln
• ln r

e

• Ihuε(x1, x2) = 1 − x1 − x2

independent of ε |uε|H1(T) → 0 for ε → 0 |Ihuε|H1(T) → 0 for ε → 0 |uε − Ihuε|L2(T) → 0 for ε → 0 Origin of this example? The estimates are valid for an interpolant based on edge mean values.

Apel Remarks on interpolation 29 / 34

Remarks on wrong conjectures II

The estimate |u − Ihu|1,T h |u|2,T ∀u ∈ H2(T) holds in 2D for isotropic and anisotropic triangles (under maximal angle condition), in 3D for isotropic tetrahedra, but not in 3D for anisotropic tetrahedra. An example is given in [Apel/Dobrowolski 92] on the basis of the example from the previous slide.

Apel Remarks on interpolation 30 / 34

Remarks on the maximal angle condition

The proof of the estimate |u − Ihu|1,T h |u|2,T ∀u ∈ H2(T) in the case of anisotropic triangles (2D) does not follow the standard arguments [Ciarlet/Raviart]. With standard arguments one would obtain an estimate like |u − Ihu|H1(T) h2

x

hy uxxL2(T) + hxuxyL2(T) + hyuyyL2(T)

hx hy

which is sharp if no maximal angle condition is assumed (hx/hy ∼ sin−1(maximal angle)): If uxxL2(T) ∼ uxyL2(T) ∼ uyyL2(T) then |u − Ihu|H1(T) h2

x

hy |u|H2(T) [Apel/Dobrowolski:92, Theorem 2].

Apel Remarks on interpolation 31 / 34

Remarks on the maximal angle condition

The proof of the estimate |u − Ihu|1,T h |u|2,T ∀u ∈ H2(T) in the case of anisotropic triangles (2D) does not follow the standard arguments [Ciarlet/Raviart]. With standard arguments one would obtain an estimate like |u − Ihu|H1(T) h2

x

hy uxxL2(T) + hxuxyL2(T) + hyuyyL2(T)

hx hy

This “new estimate” may be useful in an a posteriori context when h2

xuxxL2(T) ∼ hxhyuxyL2(T) ∼ h2 yuyyL2(T) (“alignment”) can be assured

since then |u − Ihu|H1(T) hy|u|H2(T).

Apel Remarks on interpolation 31 / 34

Remark on the coordinate system I

Estimates like |u − Ihu|H1(T) h2

x

hy uxxL2(T) + hxuxyL2(T) + hyuyyL2(T)

• r, under a maximal angle condition,

|u − Ihu|H1(T) hxuxxL2(T) + hxuxyL2(T) + hyuyyL2(T) combine the element related quantities hx and hy with basis vectors of a coordinate system (via the partial derivatives of u). The relationship of the two can be expressed in different ways:

Apel Remarks on interpolation 32 / 34

Remark on the coordinate system II

[Apel/Dobrowolski 92]: ✟✟✟✟✟ ✟❆ ❆❆ h1 h2 ✟✟✟ ❍❍❍ C [Apel/Lube 98]: | tan ϑ| h2/h1

x1 x2 ϑ h2 h1 E Apel Remarks on interpolation 33 / 34

Remark on the coordinate system II

[Apel/Dobrowolski 92]: ✟✟✟✟✟ ✟❆ ❆❆ h1 h2 ✟✟✟ ❍❍❍ C [Apel/Lube 98]: | tan ϑ| h2/h1

x1 x2 ϑ h2 h1 E

[Cao 05] considers the affine mapping x = F(ˆ x) = Bˆ x + b = UΣV T ˆ x + b with T = F(ˆ T). The columns of U form a good coordinate system [Formaggia/Perotto 01].

Apel Remarks on interpolation 33 / 34

Remark on the coordinate system II

[Apel/Dobrowolski 92]: ✟✟✟✟✟ ✟❆ ❆❆ h1 h2 ✟✟✟ ❍❍❍ C [Apel/Lube 98]: | tan ϑ| h2/h1

x1 x2 ϑ h2 h1 E

[Cao 05] considers the affine mapping x = F(ˆ x) = Bˆ x + b = UΣV T ˆ x + b with T = F(ˆ T). The columns of U form a good coordinate system [Formaggia/Perotto 01]. [Hetmaniuk/Knupp 08] give estimates where the columns of B give the coordinate directions.

Apel Remarks on interpolation 33 / 34

Summary

There are still publications on anisotropic interpolation. There is not yet an reliable and efficient a posteriori error estimator for anisotropic discretizations without any alignment condition.

Apel Remarks on interpolation 34 / 34