A posteriori error estimates in FEEC for the de Rham complex

Transcription

1 A posteriori error estimates in FEEC for the de Rham complex Alan Demlow Texas A&M University joint work with Anil Hirani University of Illinois Urbana-Champaign Partially supported by NSF DMS and a Simons Sabbatical Fellowship. A posteriori estimates for the de Rham complex p.1

2 Brief summary of FEEC Goal of Finite Element Exterior Calculus: Systematically construct and analyze stable numerical methods for PDE of Hodge-Laplace type using differential complexes and related tools such as Hodge decompositions. Characteristics in brief: Related areas: Maxwell s equations, elasticity, mixed FEM. Forebears: Hiptmair, Bossavit... Main developers: Arnold, Falk, Winther in [AFW 06, 10]. Analysis begins on abstract realization of differential complexes (Hilbert complexes). Unified analysis of Hodge-Laplace problem for all slots in complex. A posteriori estimates for the de Rham complex p.2

3 Previous work and goals Literature relevant to a posteriori estimates for FEEC: 1. MFEM scalar Laplacian: [Braess-Verfürth 96], [Carstensen 97] Maxwell s equations: [Beck et. al. 00], [Schöberl 08] This talk: [Demlow-Hirani, FoCM, 14]. Our goals: 1. Give a bird s eye view of residual a posteriori techniques and estimates for differential forms. Translate, generalize ideas from individual de Rham slots. 2. Develop a posteriori estimates for the Hodge Laplacian. Account for structure of PDE, including harmonic forms. A posteriori estimates for the de Rham complex p.3

4 The de Rham complex Definitions: Λ k (Ω) is smooth k-forms on a Lipschitz domain Ω R n. Exterior derivative d : Λ k Λ k+1 HΛ k = {v L 2 Λ k : dv L 2 Λ k+1 } de Rham complex: (, curl, div...). (H 1, H(curl), H(div)...). 0 HΛ 0 d 1 HΛ 1 d 2 dn 1 L 2 0. Codifferential (adjoint) δ : Λ k+1 Λ k d d = δ δ = 0. tr=trace operator, : Λ k Λ n k =Hodge star Note: Can also consider essential boundary conditions. ( div, curl,...). A posteriori estimates for the de Rham complex p.4

5 Hodge decomposition Hodge decomposition: HΛ k = B k H k Z k,, where: B k = range(d k 1 ). Z k is the nullspace of d k. Harmonic forms: Z k = B k H k (dim(h k ) depends on topology). Z k, is the range of δ k+1. Harmonic forms for 3D de Rham: H 0 is constants, H 3 =. k = 1, 2: H k = {p : curl p = 0, div p = 0} with appropriate BC s. A posteriori estimates for the de Rham complex p.5

6 Hodge Laplacian Basic Hodge-Laplace PDE: (δd + dδ)u = f. Mixed form: Find (σ, u, p) HΛ k 1 HΛ k H k with σ, τ dτ, u =0 (σ = δu) dσ, v + du, dv + v, p = f, v ((δd + dδ)u = f p H k ) u, q =0. (u H k ) for (τ, v, q) HΛ k 1 HΛ k H k. 3D realizations (boundary conditions vary): k = 0, 3: u = f in Ω in primal, mixed forms. k = 1, 2: (curl curl div)u = f in Ω. A posteriori estimates for the de Rham complex p.6

7 The discrete problem Approximating subspaces: T h is a regular simplicial mesh; corresponding spaces V k h HΛk (Lagrange, Nédélec, RT...) satisfy: 0 V 0 h d 0 V 1 h d 1 dn 1 V n h 0. The discrete Hodge decomposition V k h = Bk h Hk h Zk h : B k k 1 h = d(vh ) B k. H k h Zk, but H k h Hk. (But, dim(h k h ) = dim(hk ) < ). Z k h Z k,. AFW FEM: Find (σ h, u h, p h ) V k 1 h V k h Hk h satisfying σ h, τ h dτ h, u h =0, τ h V k 1 dσ h, v h + du h, dv h + v h, p h = f, v h, v h V k h, h, u h, q h =0, q h H k h. A posteriori estimates for the de Rham complex p.7

8 The Harmonic Gap Goal: Measure the effect of H k h Hk on approximation quality. Definitions: Given closed subspaces A, B of a Hilbert space W, sin (A, B) = sup x P B x, x A, x =1 gap(a, B) = max(sin (A, B), sin (B, A)). In our case: Must control gap(h k, H k h ). A posteriori estimates for the de Rham complex p.8

9 A priori analysis Lemma 1 (AFW 10). Assume there is an HΛ-bounded, commuting cochain projection Π h : V k Vh k, and let e u = u u h, etc. Then e σ HΛ k 1 + e u HΛ k + e p HΛ k inf τ Vh k 1 { + Error is bounded by σ τ HΛ + inf v V k h A best approximation term P H k h u gap(h k, H k h) inf v V k h u v HΛ + inf q V k h p q HΛ P B u v HΛ }. plus a harmonic nonconformity error (higher order...but can dominate error in some examples?). Also: Analysis can be carried out entirely at Hilbert complex level. A posteriori estimates for the de Rham complex p.9

10 Structure of a posteriori result Theorem 1. Let u h = P Z k, u h. For 0 k n, we have h e σ HΛ k 1 + e u HΛ k + e p ( ) 1/2 η 1 (K) 2 + η 0 (K) 2 + η H (p h ) 2 K T h { ( ) 1/2 + P H u h µ + µ 2 u h }. Notes: K T h η H (K, u h ) 2 Definitions of η H, η 1, η 0, µ gap(h k, H k h ) given later. Similar to a priori estimate, we have efficient and conforming residual terms + harmonic nonconformity term. Harmonic error should be higher order, but can t prove efficiency. Hilbert complex analysis not as helpful as a priori case. A posteriori estimates for the de Rham complex p.10

11 Definition of µ gap(h k, H k h ) Lemma 2. Given q i V k h, let η H (K, q i ) = h K δq i L2 (K) + h 1/2 K tr q i L2 ( K), K T h. Also, let {q i } N i=1 be an orthonormal basis for Hk h µ i = ( K T h η H (K, q i ) 2 ) 1/2. and define Then Note: N gap(h k, H k h) µ := ( µ 2 i ) 1/2. i=1 H k = {p : dp = 0, δp = 0 in Ω, tr p = 0 on Ω}. A posteriori estimates for the de Rham complex p.11

12 Definition of η 1 Interpretation: Arises from testing 1st line in MFEM. sup σ σ h, τ dτ, u u h. τ HΛ k 1, τ HΛ =1 Definition: Given K T h and 0 k n, let 0 for k = 0, h η 1 (K) = K σ h δu h K + h 1/2 K tr u h K for k = 1, h K ( δσ h K + σ h δu h K ) + h 1/2 K ( tr σ h K + tr u h K ) for 2 k n. Efficiency: η 1 (K) e u L2 Λ k (ω K ) + e σ L2 Λ k 1 (ω K ). A posteriori estimates for the de Rham complex p.12

13 Definition of η 0 Interpretation: Arises from testing second line in MFEM. sup d(σ σ h ), v + d(u u h ), dv + (p p h ), v. v HΛ k, v HΛ=1 Definition: Given K T h and 0 k n, let η 0 (K) = h K f p h δdu h K + h 1/2 K tr du h K for k = 0, f dσ h K for k = n, h K ( f dσ h p h δdu h K + δ(f dσ h p h ) K ) + h 1/2 K ( tr du h K + tr (f dσ h p h ) K ), 1 k n 1. Efficiency holds up to data oscillation. Note: f = dσ + p + δdu, and residual is f dσ h p h δdu h. More regularity of f is needed than f L 2 Λ k. A posteriori estimates for the de Rham complex p.13

14 A Hodge imbalance in our norms Question: h K δ(f dσ h p h ) K, h 1/2 K tr (f dσ h p h ) K require more regularity than f L 2. Why is this necessary? Residual: R = d(σ σ h ) + δd(u u h ) + (p p h ). dσ + p is directly approximated in L 2 by dσ h + p h δdu is only weakly approximated (in H 1 ). Must Hodge decompose f to construct error indicators with correct strength for each variable. The above indicators Hodge decompose f weakly by killing δdu. Literature: A term involving div f arises in time-harmonic Maxwell s equations if divf 0. A posteriori estimates for the de Rham complex p.14

15 Example 1: k = 0 dδ + δd = with Neumann BC s. Assumption: Standard compatibility condition Ω f = 0 holds ( f H 0 = R, and p = p h = 0). The AFW mixed method is a standard primal FEM. Estimates reduce to standard ones: u u h H 1 (Ω) ( T T h h 2 K f p h δdu h K + h K tr du h 2 K) 1/2 = ( T T h h 2 K f + u h 2 K + h K u h 2 K) 1/2. A posteriori estimates for the de Rham complex p.15

16 Example 2: k = n = 3 dδ + δd = with Dirichlet BC s. AFW gives standard mixed method with σ = u and norm H(div) L 2 (not so interesting in practice...). A posteriori estimates: η 1 = h K ( curl σ h K + σ h + u h K ) + h 1/2 K ( u h K + σ h,t K ), η 0 = f div σ h K, σ σ h H(div) + u u h L2 ( K T h η 1 (K) 2 + η 0 (K) 2 ) 1/2. Similar to [Carstensen 97], but has not appeared previously in the literature. [Ca 97] assumes convexity of Ω; no restriction here. A posteriori estimates for the de Rham complex p.16

17 Example 3: n = 3, k = 1 (Vector Laplacian) δd + dδ = (curl curl div); u n = 0, curl u n = 0 on Ω. Error indicators: Let {q i } N i=1 be an orthonormal basis for H1 h. η 1 = h K σ h + div u h + h 1/2 K u h n K, η 0 = h K ( f σ h p h curl curl u h K + curl(f dσ h p h ) K ) + h 1/2 K ( curl u h n K + (f σ h p h ) n K ), η H (K, q) = h K div q K + h 1/2 K q n K. Final estimate is exactly as in Theorem 1. First a posteriori estimates for the vector Laplacian. It seems the effect of harmonic forms on a posteriori estimates has not been studied before. A posteriori estimates for the de Rham complex p.17

18 Tool 1 for proof: Regular decompositions Need: Decomposition of v HΛ k as v = dϕ + z, where z, φ are smooth enough to give an h in interpolation estimates. Previous literature: Regular decompositions are a well-known tool for Maxwell s equations ([Hiptmair 02], [Pasciak-Zhao 02]). Generalization to arbitrary n, k: Lemma 3. Given v HΛ k, there are ϕ H 1 Λ k 1 and z H 1 Λ k such that v = dϕ + z, and ϕ H 1 + z H 1 v H. Proof uses: [Mitrea-Mitrea-Monniaux 08] for stable solution of relevant BVP, [M.-M.-Shaw 08] for bounded extension operator. A posteriori estimates for the de Rham complex p.18

19 Tool 2: Interpolation operators Desirable properties of Π h : L 2 Λ k V k h : Commutes with d, locally bounded, projection. We prove: Only local boundedness and commutativity. Lemma 4. For 0 k n, there exists Π h : L 2 Λ k Vh k dπ h = Π h d, and for K T h and z H 1 Λ k, z Π h z L2 (K) h K z H 1 (ω K ). such that We average the approaches of: [Schöberl 01, 08] constructed Π h for the 3D de Rham complex. [Christiansen-Winther 08]: Projecting, commuting, globally bounded Π h. Note: Supplanted by recent work of [Falk-Winther]? A posteriori estimates for the de Rham complex p.19

20 Thoughts on AFEM convergence in FEEC Literature: [Zhong et. al 12] prove optimality of AFEM for time-harmonic Maxwell s equations. [Chen-Holst-Xu 09] prove optimality of AFEM for controlling σ σ h L2 in case k = n, Ω simply connected. [Holst-Mihalik-Szypowski] recently extended MFEM results to arbitrary domain topology in FEEC notation. Difficulties in proving AFEM convergence for arbitrary k, natural variational norm: Lack of orthogonality (inf-sup). Harmonic errors (mess up a priori optimality). A posteriori estimates for the de Rham complex p.20

21 Convergence of AFEM for gap(h k, H k h ) Based on work in progress, can prove: Lemma 5. Assume that dim(h k ) = dim(h k h ) = 1. Then a standard AFEM based on Dörfler marking for controlling gap(h k, H k h ) using the above estimates and estimators is contractive. Notes: dim(h k ) = 1 shouldn t be essential. Computation of harmonic forms is a miniature eigenvalue problem (we know the eigenvalue, only need the eigenvectors.). AFEM convergence results for eigenvalues are harder to prove, BUT existing results require mesh fineness condition (we don t). A posteriori estimates for the de Rham complex p.21

22 A 2D Example Left: A smooth vector field u B 1 Z 1, satisfying natural BC s. Right: A (discrete) harmonic vector field q. A posteriori estimates for the de Rham complex p.22

23 Example (cont.) Computation of Hodge decomposition: Use FEM to solve z = div u on a uniform sequence of meshes. Then P B 1u = z. Expectation: (z z h ) h 2 3 if P B 1u = z has corner singularities. Iteration Energy error EOC Iteration Energy error EOC A posteriori estimates for the de Rham complex p.23

24 Example (cont.) Left: P B 1u. Right: P Z 1, u. A posteriori estimates for the de Rham complex p.24

25 Effects on error bounds Recall the harmonic nonconformity error: { P H ku gap(h k, H k h h) inf P B u v HΛ }. v Vh k Expectation: gap(h k, H k h ) h2/3, so P H k h u h 4/3. Iteration P H k h u EOC Iteration Energy error EOC P Hh u u u h L2, so u u h L2 Ch 4/3 also. Used lowest-order element, but higher order shouldn t affect rates. A (well-founded) conjecture: Mixed approximation to the Hodge Laplacian may converge suboptimally even if u, σ, p are smooth. A posteriori estimates for the de Rham complex p.25

26 Comments and Conclusions What we ve accomplished: Generalized and unified tools and techniques for a posteriori error estimation for the de Rham complex. Explained the Hodge imbalance in residuals. Gave a posteriori upper bounds for harmonic forms and their effect on errors in approximating Hodge Laplace problems. To do: Clarify whether our a posteriori estimation of the harmonic nonconformity error is efficient. Applications? Credit: Kaushik Kalyanaraman of UIUC performed some of the computations in the final section. A posteriori estimates for the de Rham complex p.26