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

Background Theoretical results Numerical experiments Nitsche-Mortaring for Singularly Perturbed Convection-Diffusion Problems Martin Schopf Joint work with Torsten Linß and Hans-G¨ org Roos Fachrichtung Mathematik Institut f¨ ur Numerische Mathematik 1. Mai 2010 1 / 26

Background Theoretical results Numerical experiments Convection-diffusion problem Consider: ✁ ε ∆ u ✁ b ☎ ∇ u � cu ✏ f in Ω , (P) u ✏ 0 on ❇ Ω , where b , c , f are smooth, 0 ➔ ε ✦ 1 and b → ♣ β 1 , β 2 q → 0 , c � 1 2 ∇ ☎ b ➙ c 0 → 0 . Outflow boundary: exponential layers of width O ♣ ε ln 1 ④ ε q 2 / 26

Background Theoretical results Numerical experiments Singular perturbation and layers Solution for ε ✏ 10 ✁ 3 3 / 26

Background Theoretical results Numerical experiments Singular perturbation and layers Oscillations in the result of standard methods ( ε ✏ 10 ✁ 3 ) 4 / 26

Background Theoretical results Numerical experiments Layer adapted discretization Shishkin-mesh, bilinear Error of bilinear Galerkin-FEM Galerkin-FEM an a 64 ✂ 64 Shishkin-mesh 5 / 26

Background Theoretical results Numerical experiments Layer adapted discretization Shishkin-mesh, linear Galerkin-FEM Error of linear Galerkin-FEM on a 64 ✂ 64 Shishkin-mesh 6 / 26

Background Theoretical results Numerical experiments Nitsche mortaring n 1 Ω 2 Ω 1 Domain decomposition: ¯ Ω ✏ ¯ Ω 1 ❨ ¯ Ω 2 Γ Interface: Γ ✏ ¯ Ω 1 ❳ ¯ Ω 2 n 2 Interface problem: ✁ ε ∆ u i ✁ b ☎ ∇ u i � cu i ✏ f i Ω i , i ✏ 1 , 2 , in u i ✏ 0 ❇ Ω ❳ ❇ Ω i , i ✏ 1 , 2 , on (IP) u 1 ✏ u 2 on Γ , ❇ u 1 � ❇ u 2 ✏ 0 Γ , on ❇ n 1 ❇ n 2 7 / 26

Background Theoretical results Numerical experiments Notation n 1 Ω 2 Ω 1 • Restriction: v i ≔ v ⑤ Ω i Γ • Broken Sobolev space: n 2 V ≔ t v P L 2 ♣ Ω q : v i P H 1 ♣ Ω i q ❅ i P t 1 , 2 ✉ , v ⑤ ❇ Ω ✏ 0 ✉ • Jump along the interface: ✈ v ✇ ≔ v 1 ⑤ Γ ✁ v 2 ⑤ Γ • Assumption: n 2 ☎ b ↕ 0 on Γ 8 / 26

Background Theoretical results Numerical experiments Weak formulation Find u P V , such that a σ M ♣ u , v q ✏ ♣ f , v q Ω ❅ v P V , where a σ M ♣ u , v q ≔ a G ♣ u , v q � a σ I ♣ u , v q , 2 2 ➳ ➳ a G ♣ u , v q ≔ ε ♣ ∇ u i , ∇ v i q Ω i ✁ ♣ b ☎ ∇ u i ✁ cu i , v i q Ω i , i ✏ 1 i ✏ 1 � � ❇ u 1 ❇ u 2 a σ I ♣ u , v q ≔ ✁ ε ✁ α 2 , ✈ v ✇ α 1 ❇ n 1 ❇ n 2 Γ � ❇ v 1 ❇ v 2 � � σ ε ✁ α 2 , ✈ u ✇ α 1 ❇ n 1 ❇ n 2 Γ ✁ ♣ b ☎ n 2 ✈ u ✇ , v 1 q Γ � ♣ γ N ✈ u ✇ , ✈ v ✇q Γ . 9 / 26

Background Theoretical results Numerical experiments Weak formulation • � ☎ , ☎ � Γ denotes duality pairing on H ✁ 1 ④ 2 1 ④ 2 ♣ Γ q ✂ H 00 ♣ Γ q 00 • α i ➙ 0, constants with α 1 � α 2 ✏ 1 ✩ ✁ 1 symmetric mortaring ✬ ✫ • σ ✏ 0 incomplete ✬ 1 nonsymmetric ✪ • γ N , positive constant (only depends on N ) � � ❇ u 1 ❇ u 2 a σ I ♣ u , v q ≔ ✁ ε ✁ α 2 , ✈ v ✇ α 1 ❇ n 1 ❇ n 2 Γ � ❇ v 1 ❇ v 2 � � σ ε ✁ α 2 , ✈ u ✇ α 1 ❇ n 1 ❇ n 2 Γ ✁ ♣ b ☎ n 2 ✈ u ✇ , v 1 q Γ � ♣ γ N ✈ u ✇ , ✈ v ✇q Γ . 10 / 26

Background Theoretical results Numerical experiments Model problem • Problem (P) on the unit square, Ω ✏ ♣ 0 , 1 q 2 • Shishkin type mesh with transition point λ ✏ ♣ λ 1 , λ 1 q and λ i ✏ 2 ε β i ln N • N , number of mesh intervals in each coordinate direction � n ��� 2 ��� ��� ��� Ω 2 ��� ��� ��� Γ ��� λ ��� Ω 1 � � ��� ��� ��� ��� � 11 / 26

Background Theoretical results Numerical experiments Discretization Find u h P V h ⑨ V , such that ❅ v h P V h a σ M ♣ u h , v h q ✏ ♣ f , v h q Ω where V h ≔ ✥ v P V : v ⑤ T P Q 1 ♣ T q ❅ T P T ♣ Ω 1 q , ✭ v ⑤ T P P 1 ♣ T q ❅ T P T ♣ Ω 2 q . N ✁ 1 ➔ H 1 , H 2 ➔ 2 N ✁ 1 “long” edge h 1 , h 2 ✏ O ♣ ε N ✁ 1 ln N q “short” edge H 2 h ¯ C 1 N ✁ 1 ➔ � ➔ C 2 N ✁ 1 � max diameter of a triangle h 2 h 1 H 1 12 / 26

Background Theoretical results Numerical experiments Properties of the discretization Galerkin orthogonality ❅ v h P V h M ♣ u ✁ u h , v h q ✏ 0 a σ Coercivity, nonsymmetric mortaring We have a � 1 M ♣ v , v q ➙ ⑦ v ⑦ 2 ❅ v P V , with the broken energy norm : 2 2 0 , Ω i �♣ γ N ✈ v ✇ , ✈ v ✇q Γ ✁ 1 ➳ ➳ ⑦ v ⑦ 2 ≔ ε ⑤ v i ⑤ 2 ⑥ v i ⑥ 2 1 , Ω i � c 0 2 ♣ b ☎ n 2 ✈ v ✇ , ✈ v ✇q Γ i ✏ 1 i ✏ 1 13 / 26

Background Theoretical results Numerical experiments Properties of the discretization Coercivity in V h Suppose ★ ε � ✁ 1 if α 1 ✏ 0 and γ N ➙ ˜ C N ♣ ln N q ✁ 1 if α 1 P ♣ 0 , 1 s . Then there exists a constant C → 0 such that ❅ v h P V h . M ♣ v h , v h q ➙ C ⑦ v h ⑦ 2 a σ 14 / 26

Background Theoretical results Numerical experiments Solution decomposition Assume that the solution u of (P) can be decomposed: u ✏ S � E 1 � E 2 � E 12 , (SD) ✞ ❇ i � j ✞ ✞ ✞ ❇ x i ❇ y j S ♣ x , y q ✞ ↕ C ✞ ✞ ✞ ✞ ❇ i � j ✞ ✞ ✞ ↕ C ε ✁ i e ✁ β 1 x ④ ε ✞ ❇ x i ❇ y j E 1 ♣ x , y q for ♣ x , y q P ¯ ✞ ✞ Ω , ✞ ✞ ❇ i � j ✞ 0 ↕ i � j ↕ 3 ✞ ↕ C ε ✁ j e ✁ β 2 y ④ ε ✞ ✞ ❇ x i ❇ y j E 2 ♣ x , y q ✞ ✞ ✞ ✞ ❇ i � j ✞ ✞ ↕ C ε ✁♣ i � j q e ✁♣ β 1 x � β 2 y q④ ε ✞ ✞ ❇ x i ❇ y j E 12 ♣ x , y q ✞ ✞ ✞ 15 / 26

Background Theoretical results Numerical experiments Interpolation error estimates Suppose ε 1 ④ 2 ↕ ♣ ln N q ✁ 2 and (SD). Then the interpolation error on our Shishkin mesh satisfies ✽ , Ω 1 ↕ C ♣ N ✁ 2 ln 2 N q , ✎ u ✁ u I ✎ ✎ ✎ ✎ ✎ u ✁ u I ✎ ✽ , Ω 2 ↕ CN ✁ 2 , ✎ ✽ , Ω 1 ↕ C ♣ N ✁ 1 ln N q , ✎ � u ✁ u I ✟✎ ✎ ∇ ε ✎ ✎ u ✁ u I ✟✎ ✽ , Ω 2 ↕ CN ✁ 1 , � ✎ ∇ ε ✎ 0 , Ω 2 ↕ CN ✁ 2 , ✎ u ✁ u I ✎ ✎ ✎ ✎ u ✁ u I ✎ 0 , Ω 1 � ✎ ✎ 0 , Ω 2 ↕ CN ✁ 1 ln N , ε 1 ④ 2 ✎ u ✁ u I ✟✎ 0 , Ω 1 � ε 1 ④ 2 ✎ u ✁ u I ✟✎ � � ✎ ∇ ✎ ∇ ✎ ✎ ✽ , Γ ↕ C ♣ N ✁ 2 ln 2 N q . ✎ ✎ ✈ u ✁ u I ✇ ✎ ✎ ✎ ✈ u ✁ u I ✇ ✎ 0 , Γ ↕ 2 ✎ ✎ 16 / 26

Background Theoretical results Numerical experiments Error analysis Supercloseness theorem Assumptions: • Solution decomposition (SD), • Triangulation T ♣ Ω 2 q shape regular with parameter � , satisfying � ✒ N ✁ 1 • Mortaring parameter γ N ✏ γ N ④ ln N , γ sufficiently large, • ⑤ a G ♣ η, ξ q⑤ ↕ CN ✁ 3 ④ 2 ln 2 N ⑦ ξ ⑦ , then ⑦ � ⑦ u I ✁ u h ⑦ u ✁ u h ⑦ ↕ ⑦ u ✁ u I ⑦ ❧♦ ♦♠♦ ♦♥ ❧♦♦♠♦♦♥ η ξ N ✁ 1 ln N � N ✁ 3 ④ 2 ln 2 N � ✟ ↕ C . 17 / 26

Background Theoretical results Numerical experiments Proof (outline) C ⑦ ξ ⑦ 2 ↕ a σ M ♣ η, ξ q ✏ a G ♣ η, ξ q � a σ I ♣ η, ξ q 2 2 ➳ ➳ ✏ ε ♣ ∇ η i , ∇ ξ i q Ω i ✁ ♣ b ☎ ∇ η i ✁ c η i , ξ i q Ω i i ✏ 1 i ✏ 1 ✂ ✡ ❇ η 1 ❇ η 2 ✁ ε ✁ α 2 , ✈ ξ ✇ ( I 1 ) α 1 ❇ n 1 ❇ n 2 Γ ✂ ❇ ξ 1 ❇ ξ 2 ✡ � σ ε ✁ α 2 , ✈ η ✇ α 1 ( I 2 ) ❇ n 1 ❇ n 2 Γ ✁ ♣ b ☎ n 2 ✈ η ✇ ,ξ 1 q Γ ( I 3 ) � ♣ γ N ✈ η ✇ , ✈ ξ ✇q Γ ( I 4 ) 18 / 26

Background Theoretical results Numerical experiments Proof (outline) ✂ ⑤ a G ♣ η, ξ q⑤ ⑦ ξ ⑦ ↕ C Galerkin part, ⑦ ξ ⑦ α 1 N ✁ 1 ln N � α 2 N ✁ 1 ✟ � γ ✁ 1 ④ 2 � from ( I 1 ) N α 1 ln 3 ④ 2 N � α 2 ε 1 ④ 2 ln 2 N � N ✁ 3 ④ 2 � ✟ from ( I 2 ) ✁ � N 1 ④ 2 ✠ � N ✁ 2 ln 2 N γ ✁ 1 ④ 2 from ( I 3 ) N ✡ N N ✁ 2 ln 2 N 1 ④ 2 � γ from ( I 4 ) ↕ CN ✁ 3 ④ 2 ln 2 N γ N ✏ γ N ④ ln N for 19 / 26

Background Theoretical results Numerical experiments Error analysis for different asymptotic behavior Supercloseness theorem Assumptions: • Solution decomposition (SD), • Triangulation T ♣ Ω 2 q shape regular with parameter � , satisfying � ✒ N ✁ 1 ln N • Mortaring parameter γ N ✏ γ N ④ ln N , γ sufficiently large, • ⑤ a G ♣ η, ξ q⑤ ↕ CN ✁ 3 ④ 2 ln 3 ④ 2 N ⑦ ξ ⑦ , then ⑦ � ⑦ u I ✁ u h ⑦ u ✁ u h ⑦ ↕ ⑦ u ✁ u I ⑦ ❧♦ ♦♠♦ ♦♥ ❧♦♦♠♦♦♥ η ξ N ✁ 1 ln N � N ✁ 3 ④ 2 ln 3 ④ 2 N � ✟ ↕ C . 20 / 26

Background Theoretical results Numerical experiments Error analysis for different asymptotic behavior Convergence theorem Assumptions: • Solution decomposition (SD), • Triangulation T ♣ Ω 2 q shape regular with parameter � , satisfying � ✒ N ✁ 3 ④ 4 ln 3 ④ 4 N • Mortaring parameter γ N ✒ N ln ✁ 3 ④ 4 N , • ⑤ a G ♣ η, ξ q⑤ ↕ CN ✁ 1 ln N ⑦ ξ ⑦ , then ⑦ u ✁ u h ⑦ ↕ CN ✁ 1 ln N . 21 / 26