Three remarks on anisotropic finite elements

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1 Three remarks on anisotropic finite elements Thomas Apel Universität der Bundeswehr München Workshop Numerical Analysis for Singularly Perturbed Problems dedicated to the 60th birthday of Martin Stynes Apel 1 / 34

2 Instead of a motivation of anisotropic finite elements Congratulations to Martin! Apel 2 / 34

3 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

4 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

5 Classical formulation Lu := div (A u) + b u + cu = f in Ω Layers due to dominating convection b: ordinary boundary layers at outflow boundary u = 0 on Γ D A u n = g on Γ N 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: [ ] ε 0 e.g. A = : diffusion small in x-direction: layers left and right 0 1 Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 5 / 34

6 Example for illustrating anisotropic diffusion div (A u) = f in Ω = (0, 1) 2 u = 0 { on Γ 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

7 Example: α = 0 div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ ] < x < 1 ε 0 A = 0 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 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ ] < x < 1 1 0 A = 0 ε Picture by G.

8 Example: α = 0.5π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ ] < x < A = 0 ε Picture by G. Winkler Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 8 / 34

9 Example: α = 0.49π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

10 Example: α = 0.48π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

11 Example: α = 0.47π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

12 Example: α = 0.46π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

13 Example: α = 0.45π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] 1 [ ] [ cos α sin α ε 0 A = sin α cos α 0 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

14 Example: α = 0.25π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ < x < ] ε 1 ε A = 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

15 Example: α = 0.05π div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ ] < x < 1 ε 0 A = 0 1 Picture by G. Winkler Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 15 / 34

16 Example: α = 0 div (A u) = f in Ω = (0, 1) 2 { 1 for 0 < x < 1 f = 2 +1 for 1 2 [ ] < x < 1 ε 0 A = 0 1 Picture by G. Winkler Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 16 / 34

17 Related results a priori error analysis (anisotropic diffusion): Li, Wheeler a posteriori error estimation (convection dominated problems): isotropic: Angermann, Kay/Silvester, Sangalli, Verfürth 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

18 SUPG discretization Let V h := {v C(Ω) : v T P 1 (T ) T T h, v = 0 on Γ D }, B h (u h, v h ) = (A u h v h ) + (b u h + cu h, v h ) + T δ T (Lu h, b v h ) T F h (v h ) = (f, v h ) + (g, v h ) ΓN + T δ T (f, b v h ) T The parameters δ T 0 satisfy δ T min{µ 2 h 2 min,t A1/ , c 0 c 2 1,T, h min,a,t A 1/2 b 1,T } where µ is the constant in some inverse inequality and h min,a,t = min h min,fa T (T ω ) with F A (x) = A 1/2 x. T Streamline upwind Petrov-Galerkin (SUPG) scheme: Find u h V h with B h (u h, v h ) = F h (v h ) v h V h. Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 18 / 34

19 Error estimation Residuals: element residual: R T := f Lu h edge residual: R E := [A u h n E ] with modification on Ω Resulting error estimator: η 2 T := α2 T r T 2 T + E T \Γ D α E β 1 E r E 2 E, η2 := T η 2 T where α T = min{c 1/2 0, h min,a,t }, β E = max T ωe (h E,T h 1 α E = α T corresponding to β E. Approximation terms: ζt 2 := α2 T R T r T 2 T + T ω T sup v H 1 Γ D (Ω)\{0} Weighted norm: u 2 := (A u, u) 2 + (c 0 u, u) Ω Dual norm: φ := φv v =: φv 1. min,a,t ), E T \Γ D α E β 1 E R E r E 2 E, ζ2 := K T h ζ 2 T. Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 19 / 34 Ω

20 Quality of the error estimator Theorem: [Apel/Nicaise/Sirch:11] The error is bounded from above by and from below by u u h m 1 (u u h, A, T h ) (η + ζ) b (u u h ) κ u u h + m 1 (v 1, A, T h )(η + ζ) where κ = max{1, c 1 0 c,t }. Alignment measure: η κ u u h + b (u u h ) + ζ m 2 1(v, A, T h ) := T h 2 min,a,t C A,T v 2 T A 1/2 v 2 Robust lower bound was obtained due to the use of the dual norm of the convective derivative, cf. also [Verfürth 05] for isotropic meshes. Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 20 / 34

21 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 on Γ with A = ( ε 0 0 1 ) ( 1, b = 0 ) Data f and g are chosen such that ( u = 10y(1 y) e x e 1+(x 1)/ε), is the exact solution.

22 Numerical test: the problem div (A u) + b u = f in Ω = (0, 1) 2 u = g on Γ with A = ( ε ) ( 1, b = 0 ) 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 L 2 (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

23 Numerical test: the discretization Mesh: piecewise uniform with an anisotropic part in the boundary strip Ω L = (1 2ε ln ε, 1) (0, 1). SUPG scheme with { εh 2 δ T = T,min in the boundary layer, elsewhere. is chosen. h 2 T,min Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 23 / 34

24 Numerical test: behaviour of the error... in the max norm and in the SUPG norm [v] 2 = B(v, v) + T T h δ T b v 2 L 2 (T ). ε = 10 4 N e L (Ω) rate [e] rate E E E E E E E E E E E E E E ε = 10 8 N e L (Ω) rate [e] rate E E E E E E E E E E E E E E Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 24 / 34

25 Numerical test: behaviour of the error estimator ε = 10 4 N η rate e rate b e rate I eff E E E E E E E E E E E E E E E E E E E E E ε = 10 8 N η rate e rate b e rate I eff E E E E E E E E E E E E E E E E E E E E E Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 25 / 34

26 Numerical test: computation of the dual norm The error in the dual norm φ is approximately computed here by Ω φ = sup φv v HΓ 1 (Ω)\{0} v sup Ω φ hv h v h V h \{0} v h D where φ h is an approximation of φ in a finite dimensional space W h, here the space of piecewise constants. By some linear algebra one can show that the latter expression is computable: Ω φ ( hv h 1/2 = φ T MK 1 M φ) T. v h sup v h V h \{0} Apel A posteriori error estimation for an anisotropic diffusion model (Collaboration with S. Nicaise and D. Sirch) 26 / 34

27 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

28 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

29 Remarks on wrong conjectures I Let T be an isotropic triangle, and I h the piecewise linear interpolant w.r.t. the vertices. Then I h u H1 (T ) u H1 (T ) u H 1 (T ) C(T ), u I h u L2 (T ) h T u H1 (T ) u H 1 (T ) C(T ). Apel Remarks on interpolation 29 / 34

30 u Remarks on wrong conjectures I Let T be an isotropic triangle, and I h the piecewise linear interpolant w.r.t. the vertices. Then I h u H1 (T ) u H1 (T ) u H 1 (T ) C(T ), u I h u L2 (T ) h T u H1 (T ) u H 1 (T ) C(T ). Counterexample: Let T be the reference triangle and r = x x 2 2. { u ε (x 1, x 2 ) = min 1, ε ln ln r } e I h u ε (x 1, x 2 ) = 1 x 1 x 2 independent of ε y x ε = 0.4 Picture by Th. Flaig u ε H 1 (T ) 0 for ε 0 I h u ε H1 (T ) 0 for ε 0 u ε I h u ε L2 (T ) 0 for ε 0 Origin of this example? Apel Remarks on interpolation 29 / 34

31 u Remarks on wrong conjectures I Let T be an isotropic triangle, and I h the piecewise linear interpolant w.r.t. the vertices. Then I h u H1 (T ) u H1 (T ) u H 1 (T ) C(T ), u I h u L2 (T ) h T u H1 (T ) u H 1 (T ) C(T ). Counterexample: Let T be the reference triangle and r = y x ε = 0.4 Picture by Th. Flaig x x 2 2. { u ε (x 1, x 2 ) = min 1, ε ln ln r } e I h u ε (x 1, x 2 ) = 1 x 1 x 2 independent of ε u ε H 1 (T ) 0 for ε 0 I h u ε H1 (T ) 0 for ε 0 u ε I h u ε 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

32 Remarks on wrong conjectures II The estimate holds u I h u 1,T h u 2,T u H 2 (T ) 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

33 Remarks on the maximal angle condition The proof of the estimate u I h u 1,T h u 2,T u H 2 (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 I h u H1 (T ) h2 x h y u xx L2 (T ) + h x u xy L2 (T ) + h y u yy L2 (T ) h y which is sharp if no maximal angle condition is assumed (h x /h y sin 1 (maximal angle)): If u xx L 2 (T ) u xy L 2 (T ) u yy L 2 (T ) then u I h u H 1 (T ) h2 x h y u H 2 (T ) [Apel/Dobrowolski:92, Theorem 2]. h x Apel Remarks on interpolation 31 / 34

34 Remarks on the maximal angle condition The proof of the estimate u I h u 1,T h u 2,T u H 2 (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 I h u H 1 (T ) h2 x h y u xx L 2 (T ) + h x u xy L 2 (T ) + h y u yy L 2 (T ) h y This new estimate may be useful in an a posteriori context when h 2 x u xx L2 (T ) h x h y u xy L2 (T ) h 2 y u yy L2 (T ) ( alignment ) can be assured since then u I h u H 1 (T ) h y u H 2 (T ). h x Apel Remarks on interpolation 31 / 34

35 Remark on the coordinate system I Estimates like u I h u H 1 (T ) h2 x h y u xx L 2 (T ) + h x u xy L 2 (T ) + h y u yy L 2 (T ) or, under a maximal angle condition, u I h u H 1 (T ) h x u xx L 2 (T ) + h x u xy L 2 (T ) + h y u yy L 2 (T ) combine the element related quantities h x and h y 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

36 Remark on the coordinate system II [Apel/Dobrowolski 92]: [Apel/Lube 98]: tan ϑ h 2 /h 1 x2 h 2 h 1 C h2 E ϑ h1 x1 Apel Remarks on interpolation 33 / 34

37 Remark on the coordinate system II [Apel/Dobrowolski 92]: [Apel/Lube 98]: tan ϑ h 2 /h 1 x2 h 2 h 1 C h2 E ϑ h1 x1 [Cao 05] considers the affine mapping x = F(ˆx) = Bˆx + b = UΣV T ˆx + b with T = F(ˆT ). The columns of U form a good coordinate system [Formaggia/Perotto 01]. Apel Remarks on interpolation 33 / 34

38 Remark on the coordinate system II [Apel/Dobrowolski 92]: [Apel/Lube 98]: tan ϑ h 2 /h 1 x2 h 2 h 1 C h2 E ϑ h1 x1 [Cao 05] considers the affine mapping x = F(ˆx) = Bˆx + b = UΣV T ˆx + b with T = F(ˆT ). 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

39 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

Goal. Robust A Posteriori Error Estimates for Stabilized Finite Element Discretizations of Non-Stationary Convection-Diffusion Problems.

Goal. Robust A Posteriori Error Estimates for Stabilized Finite Element Discretizations of Non-Stationary Convection-Diffusion Problems. Robust A Posteriori Error Estimates for Stabilized Finite Element s of Non-Stationary Convection-Diffusion Problems L. Tobiska and R. Verfürth Universität Magdeburg Ruhr-Universität Bochum www.ruhr-uni-bochum.de/num