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 A posteriori error estimation for an anisotropic diffusion model 1 (Collaboration with S. Nicaise and D. Sirch) Remarks on interpolation 2 Apel 3 / 34

Plan of the talk A posteriori error estimation for an anisotropic diffusion model 1 (Collaboration with S. Nicaise and D. Sirch) Remarks on interpolation 2 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 = 0 on Γ D A ∇ u · n = g on Γ N Layers due to dominating convection b : ordinary 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 : � 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

Example for illustrating anisotropic diffusion in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f u = 0 on Γ � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 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 . 5 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � 1 � 0 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

Example: α = 0 . 49 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 . 48 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 . 47 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 . 46 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 . 45 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � cos α � � ε � � � − sin α 0 cos α sin α A = sin α cos α 0 1 − 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 . 25 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 1 2 < x < 1 � � 1 + ε 1 − ε 1 A = 2 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 . 05 π in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 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

Example: α = 0 in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) = f � for 0 < x < 1 − 1 2 f = for 1 + 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

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 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 ) + δ T ( Lu h , b · ∇ v h ) T T � F h ( v h ) = ( f , v h ) + ( g , v h ) Γ N + δ T ( f , b · ∇ v h ) T T The parameters δ T ≥ 0 satisfy δ T ≤ min { µ − 2 h 2 min , T � A 1 / 2 � − 1 2 → 2 , 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 T ′ ⊂ ω T h min , F A ( T ′ ) with F A ( x ) = A − 1 / 2 x . and h min , A , T = min 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

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 := � � η 2 T := α 2 T � r T � 2 α E β − 1 E � r E � 2 η 2 T + E , T E ∈ ∂ T \ Γ D T where α T = min { c − 1 / 2 , h min , A , T } , β E = max T ⊂ ω E ( h E , T h − 1 min , A , T ) , 0 α E = α T corresponding to β E . Approximation terms: ζ 2 := � � � ζ 2 T := α 2 � R T ′ − r T ′ � 2 α E β − 1 E � R E − r E � 2 ζ 2 T ′ + E , T . T T ⊂ ω T E ∈ ∂ T \ Γ D K ∈T h Weighted norm: ||| u ||| 2 := ( A ∇ u , ∇ u ) 2 + ( c 0 u , u ) � Ω φ v � Dual norm: ||| φ ||| ∗ := sup ||| v ||| =: φ v 1 . v ∈ H 1 Ω Γ D (Ω) \{ 0 } 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 − 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 ) and from below by η � κ ||| u − u h ||| + ||| b · ∇ ( u − u h ) ||| ∗ + ζ where κ = max { 1 , c − 1 0 � c � ∞ , T ′ } . Alignment measure: � h − 2 min , A , T � C ⊤ A , T ∇ v � 2 T T m 2 1 ( v , A , T h ) := � A 1 / 2 ∇ v � 2 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 in Ω = ( 0 , 1 ) 2 − div ( A ∇ u ) + b · ∇ u = f u = g on Γ with � ε � � � 0 1 A = , b = 0 1 0 Data f and g are chosen such that � e − x − e − 1 +( x − 1 ) /ε � u = 10 y ( 1 − y ) , 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