Smoothed projections in finite element exterior calculus

Transcription

1 Smooted projections in finite element exterior calculus Ragnar Winter CMA, University of Oslo Norway based on joint work wit: Douglas N. Arnold, Minnesota, Ricard S. Falk, Rutgers, and Snorre H. Cristiansen, Oslo

2 Outline Wy do we need smooted projections? How do we construct smooted projections?

3 Te de Ram complex in tree dimensions Te de Ram complex: R H 1 (Ω) grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω) 0

4 Te de Ram complex in tree dimensions Te de Ram complex: R H 1 (Ω) grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω) 0 Discretizations and commuting diagrams: R H 1 (Ω) π 1 grad H(curl, Ω) π c curl H(div, Ω) π d div L 2 (Ω) 0 π 0 R W grad U curl V div Q 0.

5 Te de Ram complex in tree dimensions Te de Ram complex: R H 1 (Ω) grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω) 0 Discretizations and commuting diagrams: R H 1 (Ω) π 1 grad H(curl, Ω) π c curl H(div, Ω) π d div L 2 (Ω) 0 π 0 R W grad U curl V div Q 0. A tecnical problem: Te projections π are not defined on te entire space.

6 Te de Ram complex and differential forms By using differential forms te de Ram complex can be written as R Λ 0 d Λ 1 d d Λ n (Ω) 0. Here Λ k = C (Ω; Alt k ), were Alt k is te vector space of alternating k linear maps on R n.

7 Te de Ram complex and differential forms By using differential forms te de Ram complex can be written as R Λ 0 d Λ 1 d d Λ n (Ω) 0. Here Λ k = C (Ω; Alt k ), were Alt k is te vector space of alternating k linear maps on R n. Te exterior derivative d : Λ k Λ k+1 is defined by k+1 dω x (v 1,..., v k+1 ) = ( 1) j+1 vj ω x (v 1,..., ˆv j,..., v k+1 ), j=1 for ω Λ k and v 1,..., v k+1 R n.

8 Te de Ram complex and differential forms By using differential forms te de Ram complex can be written as R Λ 0 d Λ 1 d d Λ n (Ω) 0. Here Λ k = C (Ω; Alt k ), were Alt k is te vector space of alternating k linear maps on R n. Te exterior derivative d : Λ k Λ k+1 is defined by k+1 dω x (v 1,..., v k+1 ) = ( 1) j+1 vj ω x (v 1,..., ˆv j,..., v k+1 ), j=1 for ω Λ k and v 1,..., v k+1 R n. One easily cecks tat d 2 = 0.

9 de Ram complex: Variants of te de Ram complex 0 Λ 0 (Ω) d Λ 1 (Ω) d d Λ n (Ω) 0 were d = d k : Λ k (Ω) Λ k+1 (Ω) and d k+1 d k = 0.

10 de Ram complex: Variants of te de Ram complex 0 Λ 0 (Ω) d Λ 1 (Ω) d d Λ n (Ω) 0 were d = d k : Λ k (Ω) Λ k+1 (Ω) and d k+1 d k = 0. L 2 de Ram complex: 0 HΛ 0 (Ω) d HΛ 1 (Ω) d d HΛ n (Ω) 0 were HΛ k (Ω) = { ω L 2 Λ k (Ω) dω L 2 Λ k+1 (Ω) }

11 de Ram complex: Variants of te de Ram complex 0 Λ 0 (Ω) d Λ 1 (Ω) d d Λ n (Ω) 0 were d = d k : Λ k (Ω) Λ k+1 (Ω) and d k+1 d k = 0. L 2 de Ram complex: 0 HΛ 0 (Ω) d HΛ 1 (Ω) d d HΛ n (Ω) 0 were HΛ k (Ω) = { ω L 2 Λ k (Ω) dω L 2 Λ k+1 (Ω) } Polynomial de Ram complex 0 P r Λ 0 d Pr 1 Λ 1 d d Pr n Λ n 0

12 Coomology Te de Ram complex 0 HΛ 0 (Ω) d 0 HΛ 1 (Ω) d 1 d n 1 HΛ n (Ω) 0 is called exact if Z k = B k, were Z k := ker d k is equal to B k := range d k 1. In general, B k Z k and we assume trougout tat te kt coomology group Z k /B k is finite dimensional.

13 Coomology Te de Ram complex 0 HΛ 0 (Ω) d 0 HΛ 1 (Ω) d 1 d n 1 HΛ n (Ω) 0 is called exact if Z k = B k, were Z k := ker d k is equal to B k := range d k 1. In general, B k Z k and we assume trougout tat te kt coomology group Z k /B k is finite dimensional. Te space of armonic k-forms, H k, consists of all q Z k suc tat q, µ = 0 µ B k.

14 Coomology Te de Ram complex 0 HΛ 0 (Ω) d 0 HΛ 1 (Ω) d 1 d n 1 HΛ n (Ω) 0 is called exact if Z k = B k, were Z k := ker d k is equal to B k := range d k 1. In general, B k Z k and we assume trougout tat te kt coomology group Z k /B k is finite dimensional. Te space of armonic k-forms, H k, consists of all q Z k suc tat q, µ = 0 µ B k. Tis leads to te Hodge decomposition HΛ k (Ω) = Z k Z k = B k H k Z k.

15 Coomology Te de Ram complex 0 HΛ 0 (Ω) d 0 HΛ 1 (Ω) d 1 d n 1 HΛ n (Ω) 0 is called exact if Z k = B k, were Z k := ker d k is equal to B k := range d k 1. In general, B k Z k and we assume trougout tat te kt coomology group Z k /B k is finite dimensional. Te space of armonic k-forms, H k, consists of all q Z k suc tat q, µ = 0 µ B k. Tis leads to te Hodge decomposition HΛ k (Ω) = Z k Z k = B k H k Z k. Note tat H k = Z k /B k.

16 Hodge Laplace problem Formally: Given f Λ k, find u Λ k suc tat (d k 1 δ k 1 + δ k d k )u = f. Here δ k is a formal adjoint of d k.

17 Hodge Laplace problem Formally: Given f Λ k, find u Λ k suc tat (d k 1 δ k 1 + δ k d k )u = f. Here δ k is a formal adjoint of d k. Tis following mixed formulation is always well-posed: Given f L 2 Λ k (Ω), find σ HΛ k 1, u HΛ k and p H k suc tat

18 Hodge Laplace problem Formally: Given f Λ k, find u Λ k suc tat (d k 1 δ k 1 + δ k d k )u = f. Here δ k is a formal adjoint of d k. Tis following mixed formulation is always well-posed: Given f L 2 Λ k (Ω), find σ HΛ k 1, u HΛ k and p H k suc tat σ, τ dτ, u = 0 τ HΛ k 1 dσ, v + du, dv + p, v = f, v v HΛ k u, q = 0 q H k

19 Hodge Laplace problem Formally: Given f Λ k, find u Λ k suc tat (d k 1 δ k 1 + δ k d k )u = f. Here δ k is a formal adjoint of d k. Tis following mixed formulation is always well-posed: Given f L 2 Λ k (Ω), find σ HΛ k 1, u HΛ k and p H k suc tat σ, τ dτ, u = 0 τ HΛ k 1 dσ, v + du, dv + p, v = f, v v HΛ k u, q = 0 q H k Well-posedness follows from te Hodge decomposition and Poincaré inequality: ω L 2 c dω L 2, ω (Z k ).

20 Hodge Laplacian, Special cases: k = 0: ordinary Laplacian k = n: mixed Laplacian k = 1, n = 3: σ = div u, grad σ + curl curl u = f k = 2, n = 3: σ = curl u, curl σ grad div u = f,

21 Discretization, Abstract setting Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range.

22 Discretization, Abstract setting Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range. Hodge decomposition and Poincaré inequality follow.

23 Discretization, Abstract setting Λ k 1 d k 1 Λ k Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range. Hodge decomposition and Poincaré inequality follow. For discretization, construct a finite dimensional subcomplex.

24 Discretization, Abstract setting Λ k 1 d k 1 Λ k Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range. Hodge decomposition and Poincaré inequality follow. For discretization, construct a finite dimensional subcomplex. Define H k = (Bk ) Z k.

25 Discretization, Abstract setting Λ k 1 d k 1 Λ k Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range. Hodge decomposition and Poincaré inequality follow. For discretization, construct a finite dimensional subcomplex. Define H k = (Bk ) Z k. Discrete Hodge decomposition follows: Λ k = Bk Hk (Zk )

26 Discretization, Abstract setting Λ k 1 d k 1 Λ k Λ k 1 d k 1 Λ k Complex of Hilbert spaces wit d k bounded and closed range. Hodge decomposition and Poincaré inequality follow. For discretization, construct a finite dimensional subcomplex. Define H k = (Bk ) Z k. Discrete Hodge decomposition follows: Λ k = Bk Hk (Zk ) Galerkin s metod: Λ k 1, Λ k, H k Λ k 1, Λ k, Hk Wen is it stable?

27 Bounded cocain projections Key property: Suppose tat tere exists a bounded cocain projection. Λ k 1 d k 1 Λ k π k 1 Λ k 1 π k d k 1 Λ k π k uniformly bounded π k ω ω 0. π k a projection π k d k 1 = d k 1 π k 1

28 Bounded cocain projections Key property: Suppose tat tere exists a bounded cocain projection. Λ k 1 d k 1 Λ k π k 1 Λ k 1 Teorem π k d k 1 Λ k π k uniformly bounded π k ω ω 0. π k a projection π k d k 1 = d k 1 π k 1 If v π k v < v v Hk, ten te induced map on coomology is an isomorpism.

29 Bounded cocain projections Key property: Suppose tat tere exists a bounded cocain projection. Λ k 1 d k 1 Λ k π k 1 Λ k 1 Teorem π k d k 1 Λ k π k uniformly bounded π k ω ω 0. π k a projection π k d k 1 = d k 1 π k 1 If v π kv < v v Hk, ten te induced map on coomology is an isomorpism. gap ( H k, H k ) sup v π k v v H k, v =1

30 Bounded cocain projections Key property: Suppose tat tere exists a bounded cocain projection. Λ k 1 d k 1 Λ k π k 1 Λ k 1 Teorem π k d k 1 Λ k π k uniformly bounded π k ω ω 0. π k a projection π k d k 1 = d k 1 π k 1 If v π kv < v v Hk, ten te induced map on coomology is an isomorpism. gap ( H k, H k ) sup v π k v v H k, v =1 Te discrete Poincaré inequality olds uniformly in.

31 Bounded cocain projections Key property: Suppose tat tere exists a bounded cocain projection. Λ k 1 d k 1 Λ k π k 1 Λ k 1 Teorem π k d k 1 Λ k π k uniformly bounded π k ω ω 0. π k a projection π k d k 1 = d k 1 π k 1 If v π kv < v v Hk, ten te induced map on coomology is an isomorpism. gap ( H k, H k ) sup v π k v v H k, v =1 Te discrete Poincaré inequality olds uniformly in. Galerkin s metod is stable and convergent.

32 Proof of discrete Poincaré inequality Teorem: Tere is a positive constant c, independent of, suc tat ω c dω, ω Z k.

33 Proof of discrete Poincaré inequality Teorem: Tere is a positive constant c, independent of, suc tat ω c dω, ω Z k. Proof: Given ω Z k, define η Zk HΛ k (Ω) by dη = dω. By te Poincaré inequality, η c dω, so it is enoug to sow tat ω c η.

34 Proof of discrete Poincaré inequality Teorem: Tere is a positive constant c, independent of, suc tat ω c dω, ω Z k. Proof: Given ω Z k, define η Zk HΛ k (Ω) by dη = dω. By te Poincaré inequality, η c dω, so it is enoug to sow tat ω c η. Now, ω π η Λ k and d(ω π η) = 0, so ω π η Z k. Terefore ω 2 = ω, π η + ω, ω π η = ω, π η ω π η,

35 Proof of discrete Poincaré inequality Teorem: Tere is a positive constant c, independent of, suc tat ω c dω, ω Z k. Proof: Given ω Z k, define η Zk HΛ k (Ω) by dη = dω. By te Poincaré inequality, η c dω, so it is enoug to sow tat ω c η. Now, ω π η Λ k and d(ω π η) = 0, so ω π η Z k. Terefore ω 2 = ω, π η + ω, ω π η = ω, π η ω π η, wence ω π η. Te result follows from te uniform boundedness of π.

36 Construction of bounded cocain projections Te canonical projections, I, determined by te degrees of freedom, commute wit d. But tey are not bounded on HΛ k.

37 Construction of bounded cocain projections Te canonical projections, I, determined by te degrees of freedom, commute wit d. But tey are not bounded on HΛ k. If we apply te tree operations: extend (E) regularize (R) canonical projection (I ) we get a map Q k : HΛk (Ω) Λ k wic is bounded and commutes wit d. But it is not a projection.

38 Construction of bounded cocain projections Te canonical projections, I, determined by te degrees of freedom, commute wit d. But tey are not bounded on HΛ k. If we apply te tree operations: extend (E) regularize (R) canonical projection (I ) we get a map Q k : HΛk (Ω) Λ k wic is bounded and commutes wit d. But it is not a projection. However te composition π k = (Qk Λ k ) 1 Q k can be sown to be a bounded cocain projection.

39 Bounded projections, definitions We combine extension and smooting into an operator R ɛ : L2 Λ k (Ω) CΛ k (Ω) given by (R ɛ ω) x = ρ(y)((φ ɛy ) Eω) x dy, B 1 were Φ ɛy (x) = x + ɛg (x)y and g (x) represents te mes size.

40 Bounded projections, definitions We combine extension and smooting into an operator R ɛ : L2 Λ k (Ω) CΛ k (Ω) given by (R ɛ ω) x = ρ(y)((φ ɛy ) Eω) x dy, B 1 were Φ ɛy (x) = x + ɛg (x)y and g (x) represents te mes size. Lemma For eac ɛ > 0 sufficiently small tere is a constant c(ɛ) suc tat for all I R ɛ L(L 2 Λ k (Ω),L 2 Λ k (Ω)) c(ɛ).

41 Bounded projections, definitions We combine extension and smooting into an operator R ɛ : L2 Λ k (Ω) CΛ k (Ω) given by (R ɛ ω) x = ρ(y)((φ ɛy ) Eω) x dy, B 1 were Φ ɛy (x) = x + ɛg (x)y and g (x) represents te mes size. Lemma For eac ɛ > 0 sufficiently small tere is a constant c(ɛ) suc tat for all I R ɛ L(L 2 Λ k (Ω),L 2 Λ k (Ω)) c(ɛ). Tere is constant c, independent of ɛ and suc tat I I R ɛ L(L 2 Λ k,l2 Λ k ) cɛ.

42 Bounded cocain projections Te operator π = π k defined by π = (I R ɛ Λ k ) 1 I R ɛ, were ɛ is taken small, but not too small, as all te desired properties, i.e. π k is bounded in L(L2 Λ k (Ω), L 2 Λ k (Ω)) and L(HΛ k (Ω), HΛ k (Ω)) π k π k commutes wit te exterior derivative is a projection

43 Some references on smooted projections 1 J. Scöberl, A multilevel decomposition result in H(curl), Conference proceedings J. Scöberl, A posteriori error estimates for Maxwell equations. To appear in Mat. Comp. 3 S.H. Cristiansen, Stability of Hodge decomposition in finite element spaces of differential forms in arbitrary dimensions, Numer. Mat D.N. Arnold, R.S. Falk, R. Winter, Finite element exterior calculus, omological tecniques, and applications, Acta Numerica S.H. Cristiansen and R. Winter, Smooted projections in finite element exterior calculus. To appear in Mat. Comp.