Local coderivatives and approximation of Hodge Laplace problems - PowerPoint PPT Presentation

Local coderivatives and approximation of Hodge Laplace problems Ragnar Winther Department of Mathematics University of Oslo Norway based on joint work with Jeonghun J. Lee, UT, Austin 1

The mixed finite element method Elliptic problem: − div( K grad u ) = f in Ω , u | ∂ Ω = 0 The mixed method: � � K − 1 σ h , τ − � u h , div τ � = 0 , τ ∈ Σ h ⊂ H (div) , (1) v ∈ V h ⊂ L 2 , � div σ h , v � = � f , v � , where σ h approximates − K grad u . 2

The mixed finite element method Elliptic problem: − div( K grad u ) = f in Ω , u | ∂ Ω = 0 The mixed method: � � K − 1 σ h , τ − � u h , div τ � = 0 , τ ∈ Σ h ⊂ H (div) , (1) v ∈ V h ⊂ L 2 , � div σ h , v � = � f , v � , where σ h approximates − K grad u . In contrast to the standard finite element method the mixed method is a conservative discretization. 2

The mixed finite element method Elliptic problem: − div( K grad u ) = f in Ω , u | ∂ Ω = 0 The mixed method: � � K − 1 σ h , τ − � u h , div τ � = 0 , τ ∈ Σ h ⊂ H (div) , (1) v ∈ V h ⊂ L 2 , � div σ h , v � = � f , v � , where σ h approximates − K grad u . In contrast to the standard finite element method the mixed method is a conservative discretization. However, the method is nonlocal. 2

Properties Conservative: � � f dx = σ h · ν ds . T ∂ T 3

Properties Conservative: � � f dx = σ h · ν ds . T ∂ T Nonlocal: The map u h �→ σ h , which approximates u �→ − K grad u , defined by � � K − 1 σ h , τ = � u h , div τ � = 0 , τ ∈ Σ h is nonlocal. 3

Construction of local methods ◮ two point flux methods based on lumping of the Raviart-Thomas method, Baranger, Maitre, Oudin (1993), ◮ multipoint flux approximations, Aavatsmark et al., (1998, . . . ) ◮ various other finite volume schemes, mimetic methods ◮ discrete exterior calculus, Desbrun, Hirani, Leok, Marsden (2005) ◮ lumping of the P 1 − P 0 method, Brezzi, Fortin, Marini (2006) 4

Multipoint flux approximations The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method. 5

Multipoint flux approximations The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method. ◮ Wheeler and Yotov, A multipoint flux mixed finite element... , SIAM J. Num. Anal. (2006) ◮ Droniou and Eymard, A mixed finite volume scheme.... on any grid, Numer. Math. (2006) ◮ Klausen and W, Numer. Math. and Num. Meth. for PDE (2006) ◮ Bause, Hoffman, Knabner, Multipoint flux approximations on triangular grids, Numer. Math. (2010) ◮ Ingram, Wheeler, Yotov, A multipoint flux mixed finite element method on hexahedra, SIAM J. Num. Anal. (2010) ◮ Wheeler and Yotov, A multipoint flux mixed finite element method on distorted quadrilaterals hexahedra, Numer. Math. (2012) 5

Multipoint flux approximations The methods are constructed and described as finite volume/finite difference methods, but for convergence proofs one relies on relations to the mixed method. ◮ Wheeler and Yotov, A multipoint flux mixed finite element... , SIAM J. Num. Anal. (2006) ◮ Droniou and Eymard, A mixed finite volume scheme.... on any grid, Numer. Math. (2006) ◮ Klausen and W, Numer. Math. and Num. Meth. for PDE (2006) ◮ Bause, Hoffman, Knabner, Multipoint flux approximations on triangular grids, Numer. Math. (2010) ◮ Ingram, Wheeler, Yotov, A multipoint flux mixed finite element method on hexahedra, SIAM J. Num. Anal. (2010) ◮ Wheeler and Yotov, A multipoint flux mixed finite element method on distorted quadrilaterals hexahedra, Numer. Math. (2012) in addition to the paper by Brezzi et al. from 2006. 5

Hodge Laplace problems and their discretizations The de Rham complex: d d d H Λ 0 (Ω) → H Λ 1 (Ω) → H Λ n (Ω) − − → · · · −       � π n � π 0 � π 1 h h h d d d V 0 V 1 V n − → − → · · · − → h h h where H Λ k (Ω) = { v ∈ L 2 Λ k +1 (Ω) | dv ∈ L 2 Λ k (Ω) } , and V k h is a subspace. 6

Hodge Laplace problems and their discretizations The de Rham complex: d d d H Λ 0 (Ω) → H Λ 1 (Ω) → H Λ n (Ω) − − → · · · −       � π n � π 0 � π 1 h h h d d d V 0 V 1 V n − → − → · · · − → h h h where H Λ k (Ω) = { v ∈ L 2 Λ k +1 (Ω) | dv ∈ L 2 Λ k (Ω) } , and V k h is a subspace. We have a stable mixed methods for the Hodge Laplace problems iff there exists bounded cochain projections π h . 6

Mixed formulation of Hodge Laplace problems The non-mixed formulation: Lu = ( d ∗ d + dd ∗ ) u = f , where the unknown u is a k -form. The operator d ∗ mapping k -forms to ( k − 1)-forms can be represented in two ways, either as the L 2 adjoint of d , or as ⋆ d ⋆ , where ⋆ is Hodge star operator mapping k -forms to n − k -forms. 7

Mixed formulation of Hodge Laplace problems The non-mixed formulation: Lu = ( d ∗ d + dd ∗ ) u = f , where the unknown u is a k -form. The operator d ∗ mapping k -forms to ( k − 1)-forms can be represented in two ways, either as the L 2 adjoint of d , or as ⋆ d ⋆ , where ⋆ is Hodge star operator mapping k -forms to n − k -forms. The mixed formulation: Find ( σ, u ) ∈ H Λ k − 1 (Ω) × H Λ k (Ω) such that τ ∈ H Λ k − 1 (Ω) , � σ, τ � − � u , d τ � = 0 , v ∈ H Λ k (Ω) , � d σ, v � + � du , dv � = � f , v � , where σ = d ∗ u , and �· , ·� denotes L 2 -inner products. 7

The corresponding mixed method Find ( σ h , u h ) ∈ V k − 1 × V k h such that h τ ∈ V k − 1 � σ h , τ � − � u h , d τ � = 0 , , h v ∈ V k � d σ h , v � + � du h , dv � = � f , v � , h , h → V k − 1 where the operator d ∗ h : V k defined by h � d ∗ τ ∈ V k − 1 h u , τ � = � u , d τ � = 0 , h is nonlocal. 8

The corresponding mixed method Find ( σ h , u h ) ∈ V k − 1 × V k h such that h τ ∈ V k − 1 � σ h , τ � − � u h , d τ � = 0 , , h v ∈ V k � d σ h , v � + � du h , dv � = � f , v � , h , h → V k − 1 where the operator d ∗ h : V k defined by h � d ∗ τ ∈ V k − 1 h u , τ � = � u , d τ � = 0 , h is nonlocal. In particular, there is no natural ”discrete Hodge star operator,” ⋆ h , mapping V k h to V n − k , cf. Hiptmair, Discrete Hodge operators, h (2001). 8

Examples of Hodge Laplace problems The case k = n corresponds to mixed formulation of the scalar Laplace problem with H Λ n − 1 × H Λ n replaced by H (div) × L 2 , while the case k = 0 corresponds to the standard H 1 formulation. 9

Examples of Hodge Laplace problems The case k = n corresponds to mixed formulation of the scalar Laplace problem with H Λ n − 1 × H Λ n replaced by H (div) × L 2 , while the case k = 0 corresponds to the standard H 1 formulation. For n = 3 and k = 1: Find ( σ, u ) ∈ H 1 × H (curl) such that τ ∈ H 1 , � σ, τ � − � u , grad τ � = 0 , � grad σ, v � + � curl u , curl v � = � f , v � , v ∈ H (curl) . For n = 3 and k = 2: Find ( σ, u ) ∈ H (curl) × H (div) such that � σ, τ � − � u , curl τ � = 0 , τ ∈ H (curl) , � curl σ, v � + � div u , div v � = � f , v � , v ∈ H (div) . Both correponds to vector Laplace. 9

Abstract framework, Hilbert complexes cf. Arnold, Falk, W, Bulletin of AMS, 2010. 10

Abstract framework, Hilbert complexes cf. Arnold, Falk, W, Bulletin of AMS, 2010. We are given a structure of the form d d d V 0 → V 1 → V n − − → · · · −       � π 0 � π 1 � π n h h h d d d V 0 → V 1 → V n − − → · · · − h h h where V k = { v ∈ W k | dv ∈ W k +1 } , and V k h ⊂ V k . 10

Abstract framework, Hilbert complexes cf. Arnold, Falk, W, Bulletin of AMS, 2010. We are given a structure of the form d d d V 0 → V 1 → V n − − → · · · −       � π 0 � π 1 � π n h h h d d d V 0 → V 1 → V n − − → · · · − h h h where V k = { v ∈ W k | dv ∈ W k +1 } , and V k h ⊂ V k . The standard mixed method for the Hodge Laplace problems: Find u h ) ∈ V k − 1 × V k (˜ σ h , ˜ h such that h τ ∈ V k − 1 � ˜ σ h , τ � − � d τ, ˜ u h � = 0 , , h v ∈ V k � d ˜ σ h , v � + � d ˜ u h , dv � = � f , v � , h . where �· , ·� are inner products of the W spaces. 10

Abstract framework, Hilbert complexes cf. Arnold, Falk, W, Bulletin of AMS, 2010. We are given a structure of the form d d d V 0 → V 1 → V n − − → · · · −       � π 0 � π 1 � π n h h h d d d V 0 → V 1 → V n − − → · · · − h h h where V k = { v ∈ W k | dv ∈ W k +1 } , and V k h ⊂ V k . The standard mixed method for the Hodge Laplace problems: Find u h ) ∈ V k − 1 × V k (˜ σ h , ˜ h such that h τ ∈ V k − 1 � ˜ σ h , τ � − � d τ, ˜ u h � = 0 , , h v ∈ V k � d ˜ σ h , v � + � d ˜ u h , dv � = � f , v � , h . where �· , ·� are inner products of the W spaces. The standard theory gives optimal convergence in the V -norms for stable methods. 10