Convergence and optimality of an adaptive FEM for controlling L 2 errors

Transcription

1 Convergence and optimality of an adaptive FEM for controlling L 2 errors Alan Demlow (University of Kentucky) joint work with Rob Stevenson (University of Amsterdam) Partially supported by NSF DMS Convergence of an AFEM for L 2 errors, p.1

2 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.2

3 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.3

4 Model problem and FEM Model problem: Find u H0(Ω) 1 such that L(u, v) := u v dx = fv dx, v H0(Ω). 1 Ω Ω is a convex polygonal domain in R 2. f is piecewise polynomial (for now!). FEM: T i, i 0, is a nested sequence of regular triangulations of Ω. S i H0(Ω) 1 are Lagrange polynomials of fixed degree k on T i. u i S i satisfies L(u i, v i ) = (f, v i ), v i S i. Energy norm: v = ( Ω v 2 dx) 1/2. Ω Convergence of an AFEM for L 2 errors, p.4

5 Definition of an AFEM Standard AFEM: Iterative feedback procedure of the form Modules: Solve Estimate Mark Refine Solve for u i S i. Estimate error: Let R = u + u i = f + u i H 1 (Ω). Residual estimators bound u u i Ω = R H 1 (Ω): η 1,i (T ) 2 =h 2 T f + u i 2 L 2 (T ) + h T u i 2 L 2 ( T ), T T i, η 1,i (M) 2 = T M η 1,i (T ) 2, M T i h T = T 1/n and u i is the jump in normal derivative. Then: u u i Ω Cη 1,i (T i ). Convergence of an AFEM for L 2 errors, p.5

6 AFEM modules Standard AFEM modules (cont.): Mark subset of elements most responsible for error. Dörfler/bulk marking: For some 0 < θ < 1, M i T i chosen so η 1,i (M i ) θη 1,i (T i ). Refine: Bisect each T M i at least b 1 times, refine additionally to ensure mesh is conforming (no hanging nodes). Result: New mesh T i+1. Convergence of an AFEM for L 2 errors, p.6

7 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.7

8 Convergence of AFEM? Essential issues for AFEM convergence: 1. A priori FEM theory extracts convergence from assumption h 0. Not valid for AFEM! 2. AFEM is a highly nonlinear approximation method which seeks to optimize approximation of u over all possible refinements of T 0. Two-step answer: 1. FEM theory: AFEM converges linearly (stepwise contraction). Protagonists: [Dörfler 96], [Siebert-Nochetto-Morin 02], [Cascon et. al 08]. 2. Connect FEM theory with nonlinear approximation theory. Protagonists: [Binev-Dahmen-DeVore 04], [Stevenson 07], [Cascon et. al 08]. Convergence of an AFEM for L 2 errors, p.8

9 Convergence ingredients From [Cascon-Kreuzer-Nochetto-Siebert 08]: 1. A posteriori estimate: u u i Ω Cη 1,i (Ω). 2. Orthogonality: u u i+1 2 Ω = u u i 2 Ω u i u i+1 2 Ω. 3. Estimator reduction: For δ > 0, λ = 1 2 b/n, η 1,i+1 (Ω) 2 (1 + δ)(η i (T i ) 2 λη i (M i ) 2 ) + C(1 + 1 δ ) u i u i+1 2 Ω. 4. Contraction: For some 0 < α < 1, γ > 0, u u i+1 2 Ω + γη 1,i+1 (Ω) 2 α 2 ( u u i 2 Ω + γη 1,i (Ω) 2 ). Convergence of an AFEM for L 2 errors, p.9

10 Nonlinear approximation theory Let T be the set of all conforming meshes that are refinements of T 0 under newest-node bisection (or generalization to R 3 ). For a mesh T T, let S T be the associated FE space. Approximation classes: For s > 0, let u A1,s = sup N s N N Then u A 1,s if u A1,s <. inf T T,#T #T 0 =N inf u v. v S T If u A 1,s, then u can be approximated with rate N s : inf T T,#T #T 0 =N inf u v N s u A1,s. v S T AFEM optimality ([BDD 04], [St. 07], [CKNS 08]): If u A 1,s and θ is small enough, u u i C(#T i #T 0 ) s u A1,s. Convergence of an AFEM for L 2 errors, p.10

11 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.11

12 Data oscillation Definitions: Let P r be the L 2 projection onto the piecewise-p r functions: (P r v)q dx = vq dx, q P r. Energy-scaled data oscillation: T T osc E (T ) 2 = T T h 2 T f P k 1 f 2 L 2 (T ). Notes: If f is piecewise P k 1 on T, then osc E (T ) = 0. If f is piecewise smooth on T, then osc E (T ) = O(h k+1 ) is of higher order with respect to u u T H 1 0 (Ω) O(hk ). Convergence of an AFEM for L 2 errors, p.12

13 Cea s Lemma, AFEM, and data oscillation [CKNS 08] established that AFEM controls the total error: total error = u u i + osc E (T i ) = error + oscillation. Ceá s Lemma: The Ritz projection is stable in H 1 0(Ω) plus oscillation: u u i +osc E (T i ) C( min χ S i u χ +osc E (T i )). AFEM optimality proof of [CKNS08] uses Ceá s Lemma for the total error. Notes: Addition of the oscillation term is trivial here, but not always! Nonzero oscillation requires modification of approximation classes in optimality results. Convergence of an AFEM for L 2 errors, p.13

14 Cea s Lemma, AFEM, and data oscillation [CKNS 08] established that AFEM controls the total error: total error = u u i + osc E (T i ) = error + oscillation. Ceá s Lemma: The Ritz projection is stable in H 1 0(Ω) plus oscillation: u u i +osc E (T i ) C( min χ S i u χ +osc E (T i )). AFEM optimality proof of [CKNS08] uses Ceá s Lemma for the total error. Notes: Addition of the oscillation term is trivial here, but not always! Nonzero oscillation requires modification of approximation classes in optimality results. Convergence of an AFEM for L 2 errors, p.14

15 AFEM for other norms? AFEM for other norms (L 2, L, local energy...) behave well in practice; convergence theory is missing. Convergence results for weak norms: 1. General theory, no rates: [Morin-Siebert-Veeser 08...]. 2. Local energy (contraction result): [De., SINUM, 2010]. 3. Quasi-optimality of a slightly nonstandard L 2 AFEM: [De.-St., NumMath11] Challenges for the L 2 norm: 1. No optimal a priori estimates for shape-regular grids. 2. Ritz projection isn t stable in L 2 (No Ceá s Lemma). 3. Orthogonality? Convergence of an AFEM for L 2 errors, p.15

16 Why bother? Two downsides to studying L 2 AFEM convergence: 1. Results are only valid for convex domains because H 2 regularity is needed. 2. Any applications?? One application: L 2 norm controls pollution effects in local estimates (cf. Bank-Holst parallel adaptive algorithm). The real justification: In order to prove AFEM convergence for more interesting L p norms, it makes sense to figure out L 2 first. Convergence of an AFEM for L 2 errors, p.16

17 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.17

18 Some context: A priori estimates A priori energy estimates: If u H k+1 (Ω), u u i min χ S i u χ C h k D k+1 u L2 (Ω). Note: Energy estimate optimally reflects the local mesh size. A priori L 2 estimates: Let h = max T Ti h i. Standard duality arguments yield: Notes: u u i L2 (Ω) C h min χ S i u χ C h h k D k+1 u L2 (Ω). The L 2 estimate doesn t optimally reflect mesh grading. Optimal a priori estimates in L 2, L, W 1 don t exist for merely shape regular grids! Convergence of an AFEM for L 2 errors, p.18

19 Mildly graded meshes Mild mesh grading condition: T T is mildly graded if h T W 1 (Ω) can be constructed so that: 1. h T is uniformly equivalent to the local mesh size h T 2. h T L (Ω) is sufficiently small. Notes: L a priori estimates optimally reflecting mesh grading were proved in [Eriksson 94] under a mild mesh grading assumption. Optimal L 2 estimates for n = 1 are proved in [Brenner-Scott 08]. For n 2, mild mesh grading is the most general assumption under which optimal a priori L 2, L estimates have been proved. Definition (class of mildly graded meshes): For µ > 0, T µ = {T T : h T L (Ω) µ}. Convergence of an AFEM for L 2 errors, p.19

20 Enforcing mild mesh grading Standard conforming newest node bisection algorithm: 1. Bisect all T M i, where M i T i is the marked set. 2. Bisect additional elements to preserve conformity. Outcome: A new mesh T i+1 with #T i+1 #T i CM i. Modified algorithm to preserve mild grading: Suppose T i T µ, and we want to ensure T i+1 T µ. Then: 1. Bisect all T M i, where M i T i is the marked set. 2. Bisect enough elements to ensure local quasi-uniformity in roughly element rings about each element. 1 µ 3. Bisect additional elements to preserve conformity. Outcome: #T i+1 #T i C(µ)M i, and h i+1 µ. Convergence of an AFEM for L 2 errors, p.20

21 Example: Enforcing mild grading Left: A strongly graded mesh: h 1 2. Right: The same mesh after enforcing mild grading: h 1 8. Convergence of an AFEM for L 2 errors, p.21

22 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.22

23 Stability of the Ritz projection in L 2? [Babuška-Osborn 84]: The Galerkin projection is NOT stable in L 2. Proof: u u i L2 (Ω) C min χ S i u χ L2 (Ω). u i,h u 2 u 4 u 8 u 16 u 32 u 64 u Convergence of an AFEM for L 2 errors, p.23

24 Stability of the Ritz projection in L 2 total error Define L 2 data oscillation: osc L2 (T ) 2 = T T h 4 T f P k 1 f 2 L 2 (T ). Theorem 1 (De.-St.). Assume Ω is convex and T T µ with µ sufficiently small. Then the Ritz projection u T S T satisfies u u T L2 (Ω) C( min χ S T u χ L2 (Ω) + osc L2 (T )). The required value of µ depends on Ω, T 0, and n, but not on u. Notes: The finite element method is stable in L 2, up to data oscillation. Oscillation is typically a higher-order (O(h k+2 )) term. Proof uses residual efficiency estimate; cf. [Gudi,MCom10]. Convergence of an AFEM for L 2 errors, p.24

25 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.25

26 Orthogonality? Current AFEM convergence framework relies on orthgonality: u u i+1 2 = u u i 2 u i u i+1 2. Doesn t hold for L 2 norm! Solution: Go back to duality argument. With Ω convex, let v H 1 0(Ω) H 2 (Ω) solve v = u u i in Ω. Then with I : H 1 0(Ω) S i an interpolant, u u i 2 L 2 (Ω) = L(u u i, v Iv) C h i (u u i ) L2 (Ω) v H 2 (Ω). Proxy for the L 2 norm: We stop here and control the quantity h i (u u i ). This quantity satisfies a quasi-orthogonality relationship. Convergence of an AFEM for L 2 errors, p.26

27 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.27

28 L 2 AFEM modules Solve, Estimate, Mark, Refine: Solve for u i S i. Estimate error: Use L 2 residual estimators. η 0,i (T ) 2 =h 4 T f + u i 2 L 2 (T ) + h3/2 T u i 2 L 2 ( T ), T T i, η 0,i (M) 2 = T M η 0,i (T ) 2, M T i, u u i L2 (Ω) + h i (u u i ) Cη 0,i. Mark: For given 0 < θ < 1, M i T i chosen so η 0,i (M i ) θη 0,i (T i ). Refine: Bisect each T M i at least b 1 times, refine additionally to ensure mesh is conforming and in T µ for µ sufficiently small. Result: New mesh T i+1. Convergence of an AFEM for L 2 errors, p.28

29 L 2 Convergence ingredients Theorem 2 (De. -St.). Assume T i, T i+1 T µ. Then: 1. A posteriori estimate: h i (u u i ) Ω Cη 0,i (T i ). 2. Quasi-orthgonality: For any ɛ > 0, h i+1 (u u i+1 ) 2 Ω (1 + ɛ) h i (u u i ) 2 Ω h i+1 (u i u i+1 ) 2 Ω + Cµ2 ɛ ( u u i+1 2 L 2 (Ω) + u u i 2 L 2 (Ω) ). 3. Estimator reduction: For δ > 0, λ = 1 2 b/n, η 0,i+1 (Ω) 2 (1 + δ)(η 0,i (T i ) 2 λη 0,i (M i ) 2 ) +C(1 + 1 δ )( h i+1(u i u i+1 ) 2 + µ 2 u u i 2 L 2 (Ω) ). 4. Contraction: For some 0 < α < 1, γ > 0, µ sufficiently small: h i+1 (u u i+1 ) 2 +γη 0,i+1 (T i+1 ) 2 α 2 ( h i (u u i ) 2 +γη 0,i (T i ) 2 ). Convergence of an AFEM for L 2 errors, p.29

30 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.30

31 L 2 approximation class L 2 approximation class, with data oscillation: For s > 0, let [ ( u A0,s = sup N s inf inf u vt L2 (Ω) + osc L2 (T ) ) ]. N N T T:#T #T 0 N v T S T Let also A 0,s = {v H 1 0(Ω) : v L 2 (Ω), v As < }. Given L 2 and energy classes A 0,s0 and A 1,s1 : The energy convergence rate s 1 [ 1 n, k n ]. The L 2 convergence rate s 0 [ 1 n, k+1 n ]. If there is no data oscillation, A j,sj is characterized by Besov regularity of u. Conditions for membership in A 0, k+1 n for membership in A 1, k. n are WEAKER than those Convergence of an AFEM for L 2 errors, p.31

32 L 2 AFEM optimality Theorem 3 (De-St.). Assume that the L 2 AFEM is used with modified refine step guaranteeing that T i T µ, i 0, for µ sufficiently small. Assume also that θ is sufficiently small. Then if u A 0,s, u u i L2 (Ω) + osc L2 (T i ) C(#T i #T 0 ) s u A0,s. Here C and the required values of µ and θ depend on H 2 regularity constants, the polynomial degree k, Ω, n, and T 0. Notes: The AFEM is optimal over all of T, even though the meshes produced must be in T µ. It is hard to figure out the required value of µ (but, it probably isn t very small). Convergence of an AFEM for L 2 errors, p.32

33 Outline 1. Background and review: AFEM convergence for energy errors Definitions AFEM convergence: Contraction and optimality Data oscillation and Ceá s Lemma 2. Overcoming obstacles to proving L 2 AFEM convergence Lack of optimal estimates on shape-regular grids Instability of the Ritz projection in L 2 Lack of orthogonality 3. AFEM convergence in the L 2 norm AFEM convergence ingredients: Contraction Quasi-optimality 4. Simultaneous optimality in L 2 and H 1? Convergence of an AFEM for L 2 errors, p.33

34 Why use different AFEM for different norms? Question: If we use an AFEM with energy estimators, will we also get optimal error decrease in u u i 0 and vice-versa? Answer: Not always! Optimal energy refinement is sometimes too strong for optimal L 2 convergence, and optimal L 2 refinement is sometimes too weak for optimal energy convergence. Where we looked for a counterexample: A combination of a BVP and polynomial degree k where the L 2 AFEM achieves the generally best possible rate k+1 n, but the energy AFEM does NOT achieve its generally best possible rate k n. Convergence of an AFEM for L 2 errors, p.34

35 Example problem If u H 1 0(Ω) solves u = 1 in Ω R 3, the strongest singularity u has is u(x) r 8 7 with r the distance to the red-blue edge. Convergence of an AFEM for L 2 errors, p.35

36 Approximation classes for example problem First: We take polynomial degree k = 4, space dimension n = 3. Membership in A 1,s : u A 1,s if s k n and u Bsn+1 p,p (Ω) with p > (s ) 1. Here: u A 1, 4 requires roughly D 5 u L ɛ (Ω). Not true for singularity in example problem! Expected convergence rate for energy AFEM: s 1 = 8 7 ɛ < 4 3. Membership in A 0,s : u A 0,s if s k+1 n and u Bsn p,p(ω) with p > (s ) 1. Here: u A 0, 5 3 requires roughly D 5 u L ɛ (Ω). True! Expected rate of convergence for L 2 AFEM: s 0 = 5 3. Convergence of an AFEM for L 2 errors, p.36

37 Computational results (courtesy of ALBERTA) -2.5! -3.5! -4.5! -5.5! Slope=-1.01! L2 refinement/h1 estimator! -6.5! H1 refinement/h1 estimator! Slope=-1.12! H1 refinement/l2 estimator! -7.5! Slope=-1.47! L2 refinement/l2 estimator! slope=-1.65! -8.5! 2! 3! 4! 5! 6! 7! Refining for one norm leads to suboptimality in the other. Convergence of an AFEM for L 2 errors, p.37

38 Concluding remarks Energy case: Contraction for a standard AFEM with Dörfler marking. Quasi-optimality for θ small enough (depending on k, T 0 ). L 2 case: Contraction for modified refinement strategy enforcing T i T µ, µ small. Required value of µ depends on H 2 regularity, k, T 0. Quasi-optimality is obtained if θ is also sufficiently small. First AFEM optimality result for non-enegy norm. Extensions: Get rid of mild grading assumption? Application to optimality of AFEM for local energy errors. Other L p norms? Convergence of an AFEM for L 2 errors, p.38