Adaptive tree approximation with finite elements

Transcription

1 Adaptive tree approximation with finite elements Andreas Veeser Università degli Studi di Milano (Italy) July 2015 / Cimpa School / Mumbai

2 Outline 1 Basic notions in constructive approximation 2 Tree approximation 3 Mesh refinement with tree structure for PDEs 4 5 Convergence rates for tree approximation of gradients

3 Literature A. Veeser, Approximating gradients with piecewise polynomial functions, Found. Comp. Math. (2015). For a "generalization", covering also the L 2 -norm: F. Tantardini, A. Veeser, R. Verfürth, Localization of the best error with finite elements in the reaction-diffusion norm, Constr. Approx. (2015).

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5 H0 1 -conformity and piecewise polynomials Let M be any conforming triangulation of Ω R d and l the maximal polynomial degree. Recall P l (M) := { v K M v K P l (K ) } and take S 0 (M) = P l (M) H 1 0 (Ω) For any v P l (M) holds v H 1 0 (Ω) v C0 ( Ω) and v Ω = 0

6 Global and local best errors Write D for L 2 (D). For fixed v H1 0 (Ω), consider E ( S 0 (M) ) := inf { (v s) Ω s S 0 (M)} and, for each element K M, E ( P l (K ) ) { } := inf (v P) K P P l (K ) E ( P l (M) ) ( := E ( P l (K ) ) ) 1/2 2 K M The latter do not take into account the conformity constraints.

7 Shape coefficient For any K M, set ρ(k ) := sup { r > 0 x B r (x) K } and write σ(m) := max K M diam(k ), ρ(k ) for the shape coefficient of M.

8 Face-connecteness Assume that z K F where z is any node of S(M) := P l (M) H 1 (Ω), K is any element of M, F is any (d 1)-face of M there is a path K 0,..., K m such that K 0 = K, K m F, K i K i 1 is a (d 1)-face of M. yes no

9 Localization of best H 1 0 -error There is a constant C de depending on σ(m) and l such that E ( P l (M) ) E ( S 0 (M) ) C de E ( P l (M) ) does not hold for L 2 -norm no apriori error estimate; no regularity involved conformity constraints ok; beyond asymptotics approximately knowing the error of the Ritz projection is almost fully parallel

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11 Explicit trace and Poincaré inequalities Given an element K and a side F of K, there holds w 2 F F ( w 2 K K + 2 ) d w K diam K w K (cf. Ainsworth 07, for generalizatons cf. Veeser/Verfürth 09) If K w = 0, then (cf. Payne/Weinberger 60,... ) w K 1 π diam K w K

12 An interpolation operator Let {φ z } z N be the nodal basis functions of S(M) and define a projection onto S(M) by Πu := z N v z φ z with { P K (z) v z = F z φ zv if z interior K otherwise and P K P l (K ) st (v P K ) = E(P l (K )) and K P K = K v, F z a face containing z and sharing its type, {φ z} z as in Scott/Zhang 90

13 Sketch of proof 1 Write E ( S 0 (M) ) (v Πv) Ω and (v Πv) K (v P K ) K + (P K Πv) K and observe (Πv P K ) K φ z(v P K ) φ z K F z z K

14 Sketch of proof 2 If F z K, there holds φ z(v P K ) φ z Fz v P K Fz F z and v P K Fz C F z 1/2 v P K Fz ( 1 π ) 1/2 F z 1/2 dπ K 1/2 diam K (v P K ) K Otherwise use face-connectedness...

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16 complete Let M 0 be an admissible (in 2d: with coinciding edge labels) initial mesh of Ω and let T be the corresponding forest of master trees. Denote by M T the set of all meshes that can be generated from M 0 by simplex bisection and by M T,conf its subfamily of conforming (face-to-face) meshes. Given any (possibly non-conforming) mesh M M T, denote by complete(m ) the smallest conforming refinement of M. Thanks to third part and Remark 7.1 of Binev/DeVore 04, we have #complete(m ) #M 0 #M #M 0.

17 Mesh conformity in H 1 0 -approximation Denote by M T n and M T n,conf most n simplex bisection. the respective subfamilies with at Combining the localization of the best H0 1 -error with the inequality for complete gives: There are c 2 > 0 depending on M 0 and l such that E ( S 0 (M T,conf n ) ) C de E ( P l (M T c 2 n) )

18 Setting for tree approximation Let M 0 be an initial mesh of Ω with coinciding edge labels and T the corresponding forest of master trees. For any K T, we set ɛ(k ) = E ( v, P l (K ) ) 2 = inf (v P) 2 P P l K (K ) and, for any M M T,conf that is conforming, we have E(M) = K M ɛ(k ) E ( v, S 0 (M) ) 2.

19 Thresholding algorithm M t := for all K M 0 if ẽ(k ) > t then grow(k ) M t :=complete(m t ) where t > 0 is given and grow(k ) is (K 1, K 2 ) := bisect(k ) for i = 1, 2 if ẽ(k i ) > t then grow(k i ) else M t := M t {K i}

20 Instance optimality If #M t 3#M 0, then E ( S 0 (M t ) ) 2C de E ( P l (M T c 2 #M t ) ) for some c 2 > 0 depending on M 0 and l. Comparing with Céa Lemma, the approximant here is determined without PDE, near best in a nonlinear approximation space that, for given n, is much bigger

21 Applications The above thresholding algorithm or variants may be used to create benchmark for the adaptive solution of PDEs to coarsen in an adaptive algorithm of the form error reduction sparsity adjustment to approximate data in the PDE, e.g., in connection with with ɛ(k ) = diam(k ) 2 f 2 K (but... ) to coarsen in the adaptive solution of evolutionary PDEs (rather reaction-diffusion or L 2 -norm)

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23 Gradient conformity and best errors Note { P P P l } = { Q (P l 1 ) d i Q j = j Q i, i, j = 1,..., d } Recalling consider E ( P l (K ) ) = inf { (v P) K P P l (K ) }, E ( v, P l 1 (K ) d) L 2 := inf { v Q K Q P l 1 (K ) d}.

24 Equivalence... There holds E ( v, P l 1 (K ) d) L 2 E ( P l (K ) ) C de E ( v, P l 1 (K ) d) L 2 where C de depends only on l and d coupling of partial derivatives ok approximately knowing the error of an H 1 -seminorm-projection is almost fully parallel in terms of the error of the L 2 -projections

25 ... and about its proof Denoting by I l v the averaged Taylor polynomial of Scott/Dupont 80, E ( P l (K ) ) 2 = d d i (v I l v) 2 K = i v I l 1 ( i v) 2 K i=1 i=1 d ( i v Q i ) I l 1 ( i v Q i ) 2 K i=1 C de v Q 2 K Independence on element shape by using an idea of Dekel/Leviatan 04.