Maximum-norm a posteriori estimates for discontinuous Galerkin methods

Maximum-norm a posteriori estimates for discontinuous Galerkin methods Emmanuil Georgoulis Department of Mathematics, University of Leicester, UK Based on joint work with Alan Demlow (Kentucky, USA) DG Workshop, Ηράκλειο, Κρήτη 1

Model Problem Find u H 1 (Ω) such that u = f in Ω u = g on Ω with: Ω R d, d = 2, 3, is a polyhedral domain (not necessarily Lipschitz!). f L (Ω). g C 0 ( Ω) with extension in Wq 1 (Ω) for some q > d. 2

Overview 1 Motivation 2 Review of max norm a posteriori estimates, strategies and difficulties 3 Reliable and efficient maximum norm a posteriori bounds for dg 4 Application of these ideas to standard (conforming) FEM 5 Numerical experiments 6 Conclusions 3

Motivation Energy norm a posteriori bounds for dg well understood. Becker, Hansbo & Larson ( 03), Karakashian & Pascal ( 03, 07), Houston, Schötzau & Wihler ( 07), Ainsworth ( 07), Houston, Süli & Wihler ( 08), Ern & Stephansen ( 08), Cochez-Dhondt & Nicaise ( 08), Carstensen, Gudi & Jensen ( 08), Vohralik ( 08), Creusé & Nicaise ( 10), Ainsworth & Rankin ( 10), Bonito & Nochetto ( 10), Ern, Stephansen & Vohralik ( 10)... Controlling max norm error is of natural interest in some applications, such as obstacle problems, contact problems, etc.,... French, Larson & Nochetto ( 01), Nochetto, Siebert & Veeser ( 03, 05), Nochetto, Schmidt, Siebert & Veeser ( 06)... 4

Literature review: previous a posteriori L results for FEM Notation: u h denotes FE solution, Γ int interior skeleton, and [ u h ] the normal jump across element interfaces. 5

Literature review: previous a posteriori L results for FEM Notation: u h denotes FE solution, Γ int interior skeleton, and [ u h ] the normal jump across element interfaces. Known approaches: 1 Eriksson ( 94) 2D results (more later...) 2 Nochetto ( 95), Dari, Duran & Padra ( 00): If g = 0 and the non-degeneracy condition h max h γ min holds, then u u h L (Ω) (ln 1/h min ) α d ( h 2 (f + h u h ) L (Ω) + h u h L (Γ int)) where α 2 = 2 and α 3 = 4/3. 5

Literature review: previous a posteriori L proofs for FEM The basic approach: Respresent pointwise errors using a Green s function G, then prove regularity bounds for G in W 1 1, W 2 1,... 6

Literature review: previous a posteriori L proofs for FEM The basic approach: Respresent pointwise errors using a Green s function G, then prove regularity bounds for G in W 1 1, W 2 1,... Eriksson ( 94) (for d = 2, g = 0): Use the actual Green s function G. u(x) u h (x) =: e(x) = G(x, y) e(y)dy. Assumption: Singularities can be additively split into fundamental solution singularities and corner singularities. Ω 6

Literature review: previous a posteriori L proofs for FEM cont d Nochetto ( 95), Dari, Duran & Padra ( 00), Nochetto, Schmidt, Siebert & Veeser ( 06): Use a regularized Green s function G with regularization parameter ρ: e(x) G e dy + Cρα e C 0,α (B ρ(x)) Problem: e C 0,α (B ρ(x)) is an a priori term! Remedies: Ω 1 Nochetto ( 95), Dari, Duran & Padra ( 00) use mesh nondegeneracy. 2 Nochetto, Schmidt, Siebert & Veeser ( 06) use Hölder regularity results to bound the a priori term: e C 0,α (Ω) e W 1 q (Ω) something computable. 7

Discontinuous finite element space Consider subdivision T of Ω into triangular or quadrilateral elements, which are images of the pullbacks F κ : ˆκ κ. Define piecewise polynomial space S = {v L 2 (Ω) : v F κ P r (ˆκ)} where P r (ˆκ) = 0 i+j r 0 i,j r α ij x i y j, α ij x i y j, if ˆκ triangle if ˆκ quad Ω Γ int Ω T is a shape-regular mesh derived by systematic refinement from an initial conforming mesh. Hanging nodes are OK, with the number per face bounded uniformly. Karakashian & Pascal ( 03), Bonito & Nochetto ( 10), Ainsworth & Rankin ( 10) 8

Notation T subdivision of Ω. Consider adjacent elements κ i and κ j [u] := u i n i + u j n j scalar jump [r] := r i n i + r j n j vector jump {u} := 1 2 (u i + u j ) scalar average {r} := 1 2 (r i + r j ) vector average κ i r i u i n j κ j r j u j n i 9

Interior penalty dg method Interior penalty dg method: Find u h S s. t. B(u h, v) = l(v h ) v h S, 10

Interior penalty dg method Interior penalty dg method: Find u h S s. t. B(u h, v) = l(v h ) v h S, where, for Γ := Ω Γ int, B(u h, v h ) := h u h h v h dx Ω Γ ( ) { u h } [v h ] + { v h } [u h ] σ[u h ] [v h ] ds 10

Interior penalty dg method Interior penalty dg method: Find u h S s. t. B(u h, v) = l(v h ) v h S, where, for Γ := Ω Γ int, B(u h, v h ) := h u h h v h dx Ω l(v h ) := Ω Γ ( ) { u h } [v h ] + { v h } [u h ] σ[u h ] [v h ] ds fv dx ( v h n σv h )gds Ω For h κ := diam(κ), and for e Γ, h e := {h}, we define, for C σ > 0, σ = C σ h 1 10

Main Result Theorem Assume that Ω R d, d = 2, 3, is an arbitrary polyhedral domain. Then ( u u h L (Ω) (ln 1/h) α d h 2 (f + h u h ) L (Ω) + h[ u h ] L (Γ int) + C σ r 2( [u h ] L (Γ int) + g u h L ( Ω)) ), where α 2 = 2 and α 3 = 1, with h := min Ω h. Discussion: Results improve upon conforming FEM results in some respects. 11

Idea of proof Idea: Return to Eriksson s approach and use the standard Green s function, but with different use of regularity properties. 12

Sketch of proof Regularity estimates for Green s functions Lemma There exists a unique function G : Ω Ω R such that for each y Ω, each 1 q < d d 1, each 1 p < 4 3, and any r > 0, G(, y) W 1,q 0 (Ω) H 1 (Ω \ B r (y)) W 2,p (Ω \ B r (y)), G(, y) v dx = 0, v H0 1 (Ω), v = 0 in B r (y). Ω In addition, for any v W 1,q 0 (Ω), q > d, we have v(y) = v G(, y) dx. Ω Also, for any fixed 1 q < d d 1, G(, y) W 1,q (Ω) C, where the constant C depend on Ω but is independent of y. Finally, for (x, y) Ω Ω, x y, we have { C x y 1, d = 3, G(x, y) C ln C x y, d = 2. 13

Sketch of proof Regularity estimates for Green s functions Lemma Assume that f L p (Ω) for some 1 < p < 4 3 and that g = 0. Then there is a unique solution u W 2,p (Ω) to the model problem and u W 2,p (Ω) C p f Lp(Ω). In addition, u W 1,q 0 (Ω) for some q > d. Grisvard ( 85), Dauge ( 88, 92), Maz ya & Rossmann ( 10) 14

Sketch of proof cont d Problem: Green s function is not regular enough regarding dg face integrals... 15

Sketch of proof cont d Problem: Green s function is not regular enough regarding dg face integrals... Solution: Consider inconsistent extension B : (H 1 (Ω) + S) (H 1 (Ω) + S) R: B(w, v) := h w h v dx Ω ( ) { πv} [w] + { πw} [v] σ[w] [v] ds, Γ 15

Sketch of proof cont d We have, respectively, (u u h )(x 0 ) = u(x 0 ) u h (x 0 ) 16

Sketch of proof cont d We have, respectively, (u u h )(x 0 ) = u(x 0 ) u h (x 0 ) = u G dx g n G ds u h (x 0 ) Ω Ω 16

Sketch of proof cont d We have, respectively, (u u h )(x 0 ) = u(x 0 ) u h (x 0 ) = u G dx g n G ds u h (x 0 ) Ω Ω = l(g) B(u h, G) + B(u h, G) u h (x 0 ) 16

Sketch of proof cont d We have, respectively, (u u h )(x 0 ) = u(x 0 ) u h (x 0 ) = u G dx g n G ds u h (x 0 ) Ω Ω = l(g) B(u h, G) + B(u h, G) u h (x 0 ) = l(g πg) B(u h, G πg) + B(u h, G) u h (x 0 ) 16

Sketch of proof cont d We have, respectively, (u u h )(x 0 ) = u(x 0 ) u h (x 0 ) = u G dx Ω Ω g n G ds u h (x 0 ) = l(g) B(u h, G) + B(u h, G) u h (x 0 ) = l(g πg) B(u h, G πg) + B(u h, G) u h (x 0 ) [ ] = f (G πg) dx B(u h, G πg) + σg(g πg) ds Ω Ω [ ] + g n (G πg) ds + B(u h, G) u h (x 0 ) Ω 16

Sketch of proof cont d Term I is more or less standard, giving, ( I h 2 (f + h u h ) L (Ω) + h[ u h ] L (Γ int) + C σ r 2 [u h ] L (Γ int) ) + C σ r 2 (u h g) L ( Ω) ( ) h 2 (G πg) L1(Ω) + h 1 (G πg) L1(Ω). 17

Sketch of proof cont d Term I is more or less standard, giving, ( I h 2 (f + h u h ) L (Ω) + h[ u h ] L (Γ int) + C σ r 2 [u h ] L (Γ int) ) + C σ r 2 (u h g) L ( Ω) ( ) h 2 (G πg) L1(Ω) + h 1 (G πg) L1(Ω). The second term in the product can now be bounded using the approximation properties of π. 17

Sketch of proof cont d For term II, we separate cases: 1 elements containing or near x 0 ; 2 elements away from the element contaning x 0. 18

Sketch of proof cont d For term II, we separate cases: 1 elements containing or near x 0 ; 2 elements away from the element contaning x 0. and we use the splitting into conforming and non-conforming parts à la Karakashian & Pascal ( 03) Lemma Let T be the minimal conforming refinement of T. For every v h S, there exists a function χ W 1, (Ω) which is a piecewise polynomial on T such that v h χ L (Ω) [v h ] L (Γ int) + g v h L ( Ω). 18

Efficiency bounds Theorem Under the same assumptions, the following bounds hold: h 2 (πf + u h ) L (κ) u u h L (κ) + h 2 (f πf ) L (κ), for all κ T, and h[ u h ] L (e) u u h L (κ 1 κ 2) + h 2 (f πf ) L (κ 1 κ 2), for all faces κ 1 κ 2 =: e Γ int, for all neighbouring κ 1, κ 2 T. 19

A side benefit: Improvement for FEM New approach yields modest improvement for max-norm a posteriori estimates for conforming FEM. 20

A side benefit: Improvement for FEM New approach yields modest improvement for max-norm a posteriori estimates for conforming FEM. Corollary Assume that u h is the CG solution and that Ω R d, d = 2, 3, is an arbitrary polyhedral domain. Then u u h L (Ω) g u h L ( Ω) + C(ln 1/h min ) α d ( h 2 (f + h u h ) L (Ω) + h u h L (Γ int)), where α 2 = 2 and α 3 = 1. 20

Numerical Experiments "sol-20.gnuplot" 2D quad elements in deal.ii on an L-shaped domain: 1 0.5 0-0.5-0.20 0.4 0.6 0.81 1.2 1.4-1 -1-0.5 0 0.5 1 Figure: Adaptive mesh using cubic elements and 20 adaptive steps 21

Numerical Experiments 150 r=1 r=2 r=3 Effectivity Indices for r=1, 2, 3 Effectivity Index 100 50 0 10 2 10 3 10 4 10 5 10 6 dof 22

Numerical Experiments 10 2 10 0 10 2 L error 10 4 10 6 10 8 10 10 10 2 10 3 10 4 10 5 10 6 dof Note: No log factors in effectivity indices or observed in error reduction. 23

Numerical Experiments Ω is the non-lipschitz two-brick domain of Maz ya (or Wendland?): e 1 e 2 O We take f = 1, g = 0, with unknown exact solution. 24

Numerical Experiments: Solution properties From theory of elliptic problems on polyhedral domains (Dauge, Maz ya,...), we know: u r 2/3 e i near edges e i (i = 1, 2), where r ei =distance to e i. Known from theory. u ρ λ near O, where ρ=distance to O and λ 1.53893. Found by numerically solving a Laplace-Beltrami eigenvalue problem on a spherical cap. Effects on convergence rate: The non-lipschitz corner O has little effect on AFEM convergence rate. 25

Numerical Experiments Figure: Eigensolution on the spherical cap for O with adaptively refined mesh. 26

Numerical Experiments 10 0 L error 10 1 P2 estimator P3 estimator slope= 2/3 10 2 10 2 10 3 10 4 10 5 10 6 dof 27

Numerical Experiments Figure: Adaptive mesh using cubic elements and 4 adaptive steps 28

Numerical Experiments Figure: Adaptive mesh using cubic elements and 5 adaptive steps 29

Numerical Experiments Figure: Adaptive mesh using cubic elements and 6 adaptive steps 30

Numerical Experiments Figure: Adaptive mesh using cubic elements and 7 adaptive steps 31

Numerical Experiments Figure: Adaptive mesh using cubic elements and 8 adaptive steps 32

Outlook Conclusions: Reliable and efficiient max norm a posteriori estimates for (IP) dg; 33

Outlook Conclusions: Reliable and efficiient max norm a posteriori estimates for (IP) dg; modest improvements to the corresponding conforming FEM result; 33

Outlook Conclusions: Reliable and efficiient max norm a posteriori estimates for (IP) dg; modest improvements to the corresponding conforming FEM result; new theoretical side-developments and method of proof. 33