A Primal-Dual Weak Galerkin Finite Element Method for Second Order - PowerPoint PPT Presentation

A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang (NSF) Supported by NSF Grant DMS-1522586 Jiangsu Provincial Foundation Award BK20050538 October 26, 2015 Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Talk Outline Model Problem and Background Primal-Dual Weak Galerkin Finite Element Method Error Estimates Numerical Tests Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Model Problem—Second Order Elliptic Equations in Non-Divergence Form Find u = u ( x ) satisfying u | ∂ Ω = 0, such that d � a ij ∂ 2 ij u = f , in Ω . i , j =1 Assume a ( x ) = ( a ij ( x )) d × d ∈ L ∞ (Ω). Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Background of Second Order Elliptic Equations in Non-Divergence Form probability and stochastic processes nonlinear PDEs in conjunction with linearization techniques such as the Newton’s iterative method a ( x ) is hardly smooth nor even continuous. a ( x ) is merely essentially bounded in the application to Hamilton-Jacobi-Bellman equations. For nonlinear PDEs discretized by discontinuous finite elements, their linearization involves at most piecewise smooth coefficients. Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Variational Equation Assume the model problem has a unique strong solution in X = H 2 (Ω) ∩ H 1 0 (Ω) satisfying � u � 2 ≤ C � f � 0 . Variational Equation : Find u ∈ X such that ∀ w ∈ Y = L 2 (Ω) . b ( u , w ) = ( f , w ) b ( u , w ) = ( L u , w ) L u = � d i , j =1 a ij ∂ 2 ij u b ( · , · ) satisfies the inf-sup condition b ( v , σ ) sup ≥ Λ � σ � Y , ∀ σ ∈ Y . � v � X v ∈ X , v � =0 Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Weak Hessian W ( K ) = { v = { v 0 , v b , v g } : v 0 ∈ L 2 ( K ) , v b ∈ L 2 ( ∂ K ) , v g ∈ [ L 2 ( ∂ K )] d } . Weak Second Order Partial Derivative The weak second order partial derivative of v ∈ W ( K ) is defined as a bounded linear functional ∂ 2 ij , w v on H 2 ( K ) so that its action on each ϕ ∈ H 2 ( K ) is given by � ∂ 2 ij , w v , ϕ � K := ( v 0 , ∂ 2 ji ϕ ) K − � v b n i , ∂ j ϕ � ∂ K + � v gi , ϕ n j � ∂ K . � � weak Hessian of v ∈ W ( K ): ∇ 2 ∂ 2 w , K v = ij , w v d × d Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Discrete Weak Hessian Discrete Weak Second Order Partial Derivative A discrete weak second order partial derivative of v ∈ W ( K ), denoted by ∂ 2 ij , w , r , K v , is defined as the unique polynomial satisfying ( ∂ 2 ij , w , r , K v , ϕ ) K = ( v 0 , ∂ 2 ji ϕ ) K −� v b n i , ∂ j ϕ � ∂ K + � v gi , ϕ n j � ∂ K , ∀ ϕ ∈ P r ( K ) . discrete weak Hessian of v ∈ W ( K ): ∇ 2 w , r , K v = { ∂ 2 ij , w , r , K v } d × d Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Weak Finite Element Spaces W k ( T ) = {{ v 0 , v b , v g } ∈ P k ( T ) × P k ( e ) × [ P k − 1 ( e )] d } � � W h , k = { v 0 , v b , v g } : { v 0 , v b , v g }| T ∈ W k ( T ) , T ∈ T h W 0 h , k = {{ v 0 , v b , v g } ∈ W h , k , v b | e = 0 , e ⊂ ∂ Ω } � � S h , k = σ : σ | T ∈ S k ( T ) , T ∈ T h P k − 2 ( T ) ⊆ S k ( T ) ⊆ P k − 1 ( T ) The choice of S k ( T ) = P k − 2 ( T ) has the least degrees of freedom, but the resulting numerical solution may not be as accurate as the case of S k ( T ) = P k − 1 ( T ). Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon Figure: WG element

Primal-Dual Weak Galerkin Finite Element Algorithms Find u h ∈ W 0 h , k satisfying b h ( v , σ ) = ( f , σ ) , ∀ σ ∈ S h , k . � d i , j =1 ( a ij ∂ 2 b h ( v , σ ) = � ij , w v , σ ) T . T ∈T h The problem is not well-posed unless an inf-sup condition is satisfied. Constrained optimization problem: Find u h ∈ W 0 h , k such that � 1 � u h = argmin v ∈ W 0 2 s h ( v , v ) . h , k , b h ( v ,σ )=( f ,σ ) , ∀ σ ∈ S h , k Stablizer � h − 3 T � v 0 − v b , v 0 − v b � ∂ T + h − 1 s h ( v , v ) = T �∇ v 0 − v g , ∇ v 0 − v g � ∂ T T measures the “continuity” of v ∈ W 0 h , k in the sense that v is a classical conforming element if and only if s h ( v , v ) = 0. Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Primal-Dual Weak Galerkin Finite Element Algorithms Primal-Dual Weak Galerkin Algorithm Find ( u h ; λ h ) ∈ W 0 h , k × S h , k satisfying ∀ v ∈ W 0 s h ( u h , v ) + b h ( v , λ h ) = 0 , h , k , b h ( u h , σ ) = ( f , σ ) , ∀ σ ∈ S h , k . Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

L 2 Projections Q 0 : L 2 projection onto P k ( T ) Q b : L 2 projection onto P k ( e ) Q g = ( Q g 1 , Q g 2 , . . . , Q gd ): L 2 projection onto [ P k − 1 ( e )] d Q h : L 2 projection onto W h , k , such that on each element T , Q h w = { Q 0 w , Q b w , Q g ( ∇ w ) } Q h : L 2 projection onto S h , k Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Secon

Inf-Sup condition ( Brezzi and Babuska ) Inf-Sup Condition Assume that the coefficient matrix a = { a ij } d × d is uniformly piecewise continuous in Ω with respect to the finite element partition T h . For any σ ∈ S h , k , there exists v σ ∈ W 0 h , k satisfying b h ( v σ , σ ) ≥ 1 2 � σ � 2 0 , � v σ � 2 2 , h ≤ C � σ � 2 0 , provided that the meshsize h < h 0 for a sufficiently small, but fixed parameter h 0 > 0. Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Error Estimates Error Estimates in a Discrete H 2 Norm Assume that the coefficient functions a ij are uniformly piecewise continuous in Ω. Assume that the exact solution u is sufficiently regular such that u ∈ H k +1 (Ω). Let ( u h ; ρ h ) ∈ W 0 h , k × S h , k be primal-dual WG solution. There exists a constant C such that � u h − Q h u � 2 , h + � λ h − Q h λ � 0 ≤ Ch k − 1 � u � k +1 , provided that the meshsize h < h 0 holds true for a sufficiently small, but fixed h 0 > 0. T ∈T h � � d � v � 2 i , j =1 Q h ( a ij ∂ 2 ij v 0 ) � 2 2 , h = � T + s h ( v , v ) Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Error Estimates Error Estimates in H 1 Let u h = { u 0 , u b , u g } ∈ W 0 h , k be the primal-dual WG solution. Assume that a ij ∈ C 1 (Ω), and the exact solution u ∈ H k +1 (Ω). There exists a constant C such that 1   2  � �∇ u 0 − ∇ u � 2 ≤ Ch k � u � k +1 , T  T ∈T h provided that the meshsize h is sufficiently small and the dual problem has the H 1 -regularity with the a priori estimate � w � 1 ≤ C � θ � − 1 . Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second

Error Estimates Error Estimates in L 2 Assume a ij ∈ C 1 (Ω) ∩ Π T ∈T h W 2 , ∞ ( T ) � � . In addition, assume that the dual problem has H 2 -regularity with the a priori estimate � w � 2 ≤ C � θ � 0 , and P 1 ( T ) ⊂ S k ( T ) for all T ∈ T h . Then, we have � u 0 − u � 0 ≤ Ch k +1 � u � k +1 , � u b − Q b u � L 2 ≤ Ch k +1 � u � k +1 , � u g − Q b ∇ u � L 2 ≤ Ch k � u � k +1 . provided that the meshsize h is sufficiently small. � 1 � 1 � � � � h T � e b � 2 2 h T � e g � 2 2 � e b � L 2 = � e g � L 2 = ∂ T ∂ T T ∈T h T ∈T h Chunmei Wang Visiting Assistant Professor School of Mathematics Georgia Institute of Technology Collaborated with: Junping Wang A Primal-Dual Weak Galerkin Finite Element Method for Second