Finite Element Exterior Calculus and Applications Part V

Transcription

1 Finite Element Exterior Calculus and Applications Part V Douglas N. Arnold, University of Minnesota Peking University/BICMR August 15 18, 2015

2 De Rham complex 0 H 1 (Ω) grad H(curl, Ω) curl H(div, Ω) div L 2 (Ω) 0 0 H 1 (Ω) grad L 2 (Ω; R 3 ): standard formulation of scalar Laplacian H 1 (Ω) grad H(curl, Ω) curl L 2 (Ω; R 3 ): 1-form Laplacian, Maxwell s equation based on E and σ = div ɛe = 0 H(curl, Ω) curl H(div, Ω) div L 2 (Ω): 2-form Laplacian, Maxwell s equation based on B and E H(div, Ω) div L 2 (Ω) 0: mixed formulation of scalar Laplacian 1 / 20

3 De Rham complex in 2D 0 H 1 (Ω) grad H(rot, Ω) rot L 2 (Ω) 0 or 0 H 1 (Ω) curl H(div, Ω) div L 2 (Ω) 0 0 H 1 (Ω) curl L 2 (Ω; R 2 ): standard formulation of scalar Laplacian H 1 (Ω) curl H(div, Ω) div L 2 (Ω): 1-form Laplacian H(div, Ω) div L 2 (Ω) 0: mixed formulation of scalar Laplacian (Darcy flow) 2 / 20

4 Stokes complex in 2D and 3D 0 H 2 (Ω) curl H 1 (Ω; R 2 ) div L 2 (Ω) 0 Falk-Neilan shape fns: P 5 Λ 0 / P 4 Λ 1 / P 3 Λ 2 0 H 1 (Ω) div H 1 (Ω, curl; R 3 ) curl H 1 (Ω; R 3 ) div L 2 (Ω) 0 J. Evans 11 3 / 20

5 Elasticity complex 0 H 1 (Ω; R 3 ) sym grad curl T curl H(curl T curl, Ω) H(div, Ω; S 3 3 ) div L 2 (Ω; R 3 ) 0 H 2 (Ω) curl curl H(div, Ω; S 2 2 ) div L 2 (Ω; R 2 ) 0 0 H 1 (Ω; R 2 ) sym grad H(rot rot, Ω; S 2 2 rot rot ) L 2 (Ω) 0 0 H 1 (Ω; R 2 sym grad ) H(rot rot, Ω; S 2 2 ): displacement formulation of elasticity H((div, Ω; S 2 2 ) div L 2 (Ω; R 2 ) 0: mixed formulation of elasticity (strong symmetry) 4 / 20

6 Mixed elasticity elements (2D strong symmetry) 0 H 2 (Ω) curl curl H(div, Ω; S 2 2 ) div L 2 (Ω; R 2 ) 0 AW Hu Jun-Shangyou Zhang / 20

7 New complexes from old: a simple case Suppose 0 W 1 d, V 1 W 2 and 0 W 1 d,ṽ 1 W 2 are closed Hilbert complexes, and that there is a bounded linear isomorphism S : W 1 W 2. d, V 1 0 W 1 W 2 d, Ṽ 1 0 W 1 W 2 S We define a new short Hilbert complex: V 1 = {(u, φ) V 1 Ṽ 1 du = Sφ} W 1 is the competion of V 1 wrt the norm u, φ W := u W W 2 = W 2 d : V 1 W 1 W 2 is given by d(u, φ) = dφ. (S is injective) THEOREM Suppose that the initial two H-complexes are closed and exact. Then 0 W 1 d,v1 W 2 is also a closed, exact H-complex. 6 / 20

8 The Hodge Laplacian for the derived complex d, V 1 0 W 1 W 2 d, Ṽ 1 0 W 1 W 2 S 0 W 1 d,v1 W 2 Hodge Lap: Find (u, φ) V 1 st dφ, dψ = f, v, (v, ψ) V 1 V 1 = {(u, φ) V 1 Ṽ 1 du = Sφ} = {(u, φ) V 1 Ṽ 1 du Sφ, µ = 0 µ W 2 } Implement via Lagrange multiplier: Find u V 1, φ Ṽ 1, λ W 2 s.t. dφ, dψ + λ, dv Sψ = f, v, v V 1, ψ Ṽ 1, du Sφ, µ = 0, µ W 2 New norm on W 2 µ, dv Sψ : µ = sup (SṼ v v V 1,ψ Ṽ 1 V + 1 is dense) ψ Ṽ This mixed method satisfies the Brezzi conditions and so is well-posed. 7 / 20

9 Discretization The idea is to mimic the construction on the discrete level. Choose two discrete subcomplexes which admit commuting projections: 0 V 1 h d V 2 h 0 Ṽ 1 h d Ṽ 2 h Create a discrete connection map as S h = Π 2 h S : Ṽ1 h V 2 h where Π 2 h is the canonical projection. This gives the mixed method: Find u h V 1 h, φ h Ṽ 1 h, λ h V 2 h s.t. dφ h, dψ + λ h, dv Π 2 h Sψ = f, v, v V 1 h, ψ Ṽ1 h, du h Π 2 h Sφ h, µ = 0, µ V 2 h We make the surjectivity assumption Π 2 h S Π 1 h = Π 2 h. We can then prove that the mixed method is convergent. 8 / 20

10 Example: the biharmonic grad, H 1 0 L 2 L 2 (Ω; R n ) grad, H 1 0 L 2 (Ω; R n ) L 2 (Ω; R n n ) I V 1 = {(u, φ) H 1 (Ω) H 1 (Ω; R n ) φ = grad u} = {(u, grad u) u H 2 (Ω)} = H 2 d, V 1 0 W 1 W 2 grad grad, H 2 0 L 2 L 2 (Ω; R n n ) 9 / 20

11 FEEC discretization of the biharmonic grad, H 1 0 L2 0 L2 (Ω; Rn ) L2 (Ω; Rn ) I grad, H 1 L2 (Ω; Rn n ) grad, H 1 0 I 0 Rn grad, H 1 Rn This gives a family of mixed methods for the biharmonic based on a different formulation than the classical methods (Ciarlet Raviart, Hellan Herman Johnson,... ). It is related (in 2D) to the approach of Dura n Liberman for the Reissner Mindlin plate. 10 / 20

12 Elasticity with weak symmetry The mixed formulation of elasticity with weak symmetry is more amenable to discretization than the standard mixed formulation. Fraeijs de Veubeke 75 p = skw grad u, Aσ = grad u p Find σ L 2 (Ω; R n n ), u L 2 (Ω; R n ), p L 2 (Ω; R n n skw ) s.t. Aσ, τ + u, div τ + p, τ = 0, τ L 2 (Ω; R n n ) div σ, v = f, v, v L 2 (Ω; R n ) σ, q = 0, q L 2 (Ω; R n n skw ) This is exactly the mixed Hodge Laplacian for the complex L 2 A (Ω; Rn n ) supposing that it is exact. ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 11 / 20

13 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S L 2 (Ω; R n R n n skw ) skw div L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 Sτ = τ T tr(τ)i (invertible) 12 / 20

14 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S L 2 (Ω; R n R n n skw ) skw div q L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 v Sτ = τ T tr(τ)i (invertible) 12 / 20

15 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S L 2 (Ω; R n R n n skw ) skw div q L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 ρ v Sτ = τ T tr(τ)i (invertible) 12 / 20

16 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S L 2 (Ω; R n R n n skw ) skw div q skw ρ L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 ρ v Sτ = τ T tr(τ)i (invertible) 12 / 20

17 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S ψ L 2 (Ω; R n R n n skw ) skw div q skw ρ L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 ρ v Sτ = τ T tr(τ)i (invertible) 12 / 20

18 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S ψ L 2 (Ω; R n R n n skw ) skw div q skw ρ L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 φ ρ v Sτ = τ T tr(τ)i (invertible) 12 / 20

19 Well-posedness L 2 A (Ω; Rn n ) ( div, skw) L 2 (Ω; R n ) L 2 (Ω; R n n skw ) 0 To show the complex is exactness, and so the system is well-posed, we relate it to two de Rham complexes with commuting connecting maps: S ψ L 2 (Ω; R n R n n skw ) skw div q skw ρ L2 (Ω; R n n skw ) 0 curl div L 2 (Ω; R n n ) L 2 (Ω; R n n ) L 2 (Ω; R n ) 0 φ curl φ+ ρ v Sτ = τ T tr(τ)i (invertible) 12 / 20

20 Discretization To discretize we select discrete de Rham subcomplexes with commuting projs V 1 h div V 2 h 0, Ṽ0 h curl Ṽh 1 div Ṽh 2 0 to get the discrete complex Ṽ 1 h Rn ( div, π2 h skw) (Ṽ 2 h Rn ) ( V 2 h Rn n skw ) 0 We get stability if we can carry out the diagram chase on: π 1 h S curl V 1 h Rn n skw V 2 h Rn n skw 0 Ṽ 0 h Rn Ṽ 1 h Rn Ṽ 2 h Rn 0 div π 2 h skw div This requires that π 1 h S : Ṽ0 h Rn V 1 h Rn n skw is surjective. 13 / 20

21 Stable elements The requirement that π h 1S : Ṽ0 h Rn V h 1 Rn n skw can be checked by looking at DOFs. is surjective The simplest choice is P r Λ n 1 div P r Λ n 0, P r+1 Λn 2 curl P r Λ n 1 div P r 1 Λ n 0 This gives the elements of DNA Falk Winther 07 σ u p Other elements: Cockburn Gopalakrishnan Guzmán, Gopalakrishnan Guzmán, Stenberg, / 20

22 Nearly incompressible material displacement mixed 15 / 20

23 Einstein Bianchi equations Riem = Ricci + Weyl ( ) E B Weyl = (C abcd ) = B E E, B 3 3 symmetric, traceless Einstein equations + Bianchi identity = Einstein Bianchi eqs: Find: E, B : [0, T] S 3 3 such that Ė = curl B, Ḃ = curl E, div E = 0, div B = 0, tr E = 0, tr B = / 20

24 Einstein Bianchi as an abstract Hodge wave equation L 2 (Ω) grad grad,h2 L 2 (Ω; S) curl,h(curl) L 2 (Ω; T) Find (σ, E, B) : [0, T] H 2 H(curl; S) L 2 (Ω; T) s.t. σ, τ u, grad grad τ = 0, τ H 2, Ė, F + grad grad σ, F + B, curl F = 0, F H(curl; S), Ḃ, C curl E, C = 0, C L 2 (Ω; T). σ = div div E, Ė = grad grad σ sym curl B, Ḃ = curl E THEOREM Suppose σ(0) = 0 and E(0) and B(0) are TSD. Then σ = 0 and E and B are TSD for all time, and E and B satisfy the linearized EB equations. 17 / 20

25 Obstacles to discretization To proceed we need finite element subspaces which form a subcomplex with bounded cochain projections. There are two serious obstacles. 1. It is difficult to create a finite element subspace of H 2 because of the second derivatives. 2. It is difficult to create a finite element subspace of H(curl; S) because of the symmetry. For each of these obstacles we are guided by their solution in simpler context (biharmonic, elasticity). 18 / 20

26 The FEEC formulation of the EB system Combining these ideas leads to a first order formulation of EB using six variables. L2 (Ω) grad curl L2 (Ω; R3 ) I skw grad curl L2 (Ω; R3 ) L2 (Ω; R3 3 ) L2 (Ω; R3 3 ) E B FEEC guides us to an appropriate choice of elements. L2 (Ω; R3 ) grad Π1 curl Π2 skw grad R3 R3 curl R3 19 / 20

27 Which complexes can we construct from the de Rham complex? 0 L 2 grad L 2 R 3 curl L 2 R 3 div L 2 0 id skw tr 0 L 2 R 3 grad L 2 R 3 3 curl L 2 R 3 3 div L 2 R 3 0 inc S skw 0 L 2 R 3 grad L 2 R 3 3 curl L 2 R 3 3 div L 2 R 3 0f I inc id 0 L 2 grad L 2 R 3 curl L 2 R 3 div L 2 0 Diagram commutes. Diagonal maps are isomorphisms, subdiagonal injections, superdiagonal surjections. 20 / 20