Polynomial-degree-robust liftings for potentials and fluxes

Transcription

1 Polynomial-degree-robust liftings for potentials and fluxes and Martin Vohraĺık Université Paris-Est, CERMICS (ENPC) and SERENA Project-Team, INRIA, France RAFEM, Hong-Kong, March 2017

2 Outline 1. Motivation: a posteriori error estimates 2. Main results on H 1 and H(div) polynomial liftings 3. Main ingredients of proofs 4. Numerical results (with V. Dolejší) 5. Global H(div) polynomial liftings (with I. Smears)

3 A posteriori error estimates Model problem in Ω R d with data f L 2 (Ω) u H 1 0 (Ω) s.t. ( u, v) Ω = (f, v) Ω, v H 1 0 (Ω) H 1 0 -conforming FEM solution u h on a simplicial mesh T h Fully computable error upper bound (u u h ) 2 Ω K T h η 2 K Local error indicators η K represent local error lower bounds up to data oscillation η K C eff (u u h ) ωk + osc(f, ω K ) Pioneered by [Babuška, Rheinbolt 78]; recent textbook [Verfürth 13] classical technique to compute the ηk s is residual-based drawback: upper bound features an undetermined constant

4 Equilibrated flux reconstruction Exact flux σ := u H(div, Ω) s.t. σ = f (equilibrium) An equilibrated flux reconstruction is s.t. σ h H(div, Ω) ( σ h, 1) K = (f, 1) K, K T h Setting η F,K := u h + σ h K and η osc,k := h K π f σ h K, (u u h ) 2 Ω K T h (η F,K + η osc,k }{{} =η K Literature: hypercircle method [Prager, Synge 47]; computational mechanics [Ladevèze et al. 75]; textbooks [Ainsworth, Oden 00; Repin 08] ) 2

5 A simple proof Residual ρ(u h ) H 1 (Ω) s.t. ρ(u h ), ϕ Ω := (f, ϕ) Ω ( u h, ϕ) Ω, ϕ H 1 0 (Ω) (u u h ) Ω = ρ(u h ) H 1 (Ω) = Introduce H(div, Ω)-flux and use Green s formula sup ρ(u h ), ϕ Ω ϕ H0 1(Ω); ϕ Ω=1 ρ(u h ), ϕ Ω = (f σ h, ϕ) Ω ( u h + σ h, ϕ) Ω Cauchy Schwarz and Poincaré Steklov inequalities (equilibration) (f σ h, ϕ) Ω = K T h (f σ h, ϕ ϕ K ) K K T h h K π f σ h K ϕ K ( u h + σ h, ϕ) Ω K T h u h + σ h K ϕ K Poincaré (1894) [eigenvalue pb], Steklov (1897) [d = 1], Payne, Weinberger (60) [d = 2], Bebendorf (03) [d 3]

6 a5 a4 a2 a3 How to build σ h? Global flux equilibration (... expensive) σ h := arg min u h + v h Ω v h V h, v h =Π ( Vh )f for some global Raviart Thomas spaces V h H(div, Ω) Cheap local flux equilibration by working on FE stars Ta T h : patch of elements sharing vertex a V h ; domain ω a a Vh a1 patch ωa ψa(a) =1,ψa(a ) =0 Literature on a posteriori analysis on FE stars: [Babuška, Miller 87; Morin, Nochetto, Siebert 03; Cohen, DeVore, Nochetto 12; Veeser, Verfürth 12]

7 Local flux equilibration on FE stars Hat basis functions {ψ a } a Vh with local PU a V K ψ a K = 1 We focus for simplicity on FE stars around interior vertices flux equilibration for boundary vertices depends on the BC s for model problem, see [Dolejší, AE, MV, 16] Raviart Thomas space V a h H 0(div, ω a ) (Neumann BC s on ω a ) σ a h := arg min ψ a u h + v h ωa v h Vh a, v h=gh a σ h := a V h σ a h with data gh a := Π ( Vh a) (f ψ a u h ψ a ) a V K gh a = Π ( Vh a) f (PU), (gh a, 1) ω a = 0 (Galerkin s orthogonality on hat basis functions) Literature [Destuynder, Métivet 99; Braess, Schöberl 08; MV 08; AE, MV 10]

8 p-robust local efficiency Let p be the polynomial degree used to compute u h Local flux equilibration using RT spaces of order p Key result: Local lower error bound (ω K = a VK ω a ) η F,K = u h + σ h K C eff (u u h ) ωk + osc(f, ω K ) C eff depends on patch geometry (mesh regularity) Ceff is p-robust for d = 2 [Braess, Pillwein, Schöberl 09] Ceff is p-robust for d = 3 [AE, MV 16]... in contrast to residual-based estimators [Melenk, Wohlmuth 01]

9 A two-step proof Two-step proof using -dimensional, local problems replaces classical bubble-function argument by Verfürth see [AE, MV 15] We have C eff = C st C PS Cst results from p-robust stability properties of RT spaces [BPS 09] CPS results from Poincaré Steklov inequalities on FE stars [Carstensen, Funken 00], values estimated in [Veeser, Verfürth 12]

10 Step 1 Set τ a h := ψ a u h and V a h H 0(div, ω a ) = V a Local PU yields u h + σ h K a V K τ a h + σa h ω a Recall that local flux equilibration delivers σ a h := arg min τ v h Vh a, v h a + v h ωa h=gh a Local -dimensional constrained min. pb. with polynomial data σ a := arg min τh a + v ωa v V a, v=gh a Thm. The following (nontrivial) result holds: τh a + σh a ωa C st τh a + σ a ωa Note the (trivial) converse bound τ a h + σa ωa τ a h + σa h ωa

11 Step 2 Evaluate τh a + σa ωa using equivalent problem in primal form r a H (ω 1 a) = {v H 1 (ω a) (v, 1) ωa = 0} s.t. ( r a, v) ωa = (τh a, v) ωa + (gh a, v) ωa v H (ω 1 a) Equivalence of primal/dual energies τ a h + σ a ωa = min τ a v V a, v=g h a h + v ωa = r a ωa Since ( r a, v) ωa = ( (u u h ), (ψ av)) ωa + osc(f, ω a) (ψav) ωa (1 + C PS,ωa h ωa ψ a L (ω a)) v ωa we can conclude that τh a + σ a ωa C PS (u u h ) ωa + osc(f, ω a ) with C PS = max a Vh (1 + C PS,ωa h ωa ψ a L (ω a))

12 Nonconforming case (u h H 1 0 (Ω)) Error measure w.r.t. broken gradient T of discrete solution Additional nonconformity estimator in error upper bound H 1 0 -potential reconstruction s h H 1 0 (Ω) Setting η NC,K := T (u h s h ), T (u u h ) 2 ( ) 2 ( ) 2 ηf,k + η osc,k + ηnc,k K T h K T h Typically, s h is built by prescribing its nodal values as averages see [Achdou, Bernardi, Coquel 03; Karakashian, Pascal, 03] for Crouzeix Raviart and IPDG residual-based estimates... not p-robust [Burman, AE 07; Houston, Schötzau, Wihler 07]

13 Potential reconstruction Local FEM solves of order (p + 1) on vertex-based patches s a h P p+1 (T a) H0 1 (ω a) set sh = a V h sh a p-robust local efficiency proved in [AE, MV 15] in 2D assuming [[uh ]], 1 F = 0, η NC,K C eff T (u u h ) ωa, C eff = C stc bps a V K two-step proof as above (no oscillation here) 1st step: mixed RT solve of order p for rotated gradient of s a h 2nd step: broken PS inequalities, see also [Carstensen, Merdon 13] jump seminorm added to error and estimator if [[uh ]], 1 F 0 use discrete gradient for DG (instead of broken gdt); broken PS and jump seminorm can be avoided for symmetric IPDG [AE, MV 15]

14 Main results [AE, MV 16] 3D, p-robust flux and potential reconstructions Thm. 1 p-robust, H 1 -stable polynomial lifting for potentials Thm. 2 p-robust, H(div)-stable polynomial lifting for fluxes Our proofs are constructive (as 2D proofs) e.g., build σ a h Vh a s.t. σ h a = gh a, τh a + σ h a ωa C st τh a + σ a ωa then τ a h + σh a ωa τh a + σ h a ωa C st τh a + σ a ωa Main challenges how to enumerate tetrahedra in 3D star (triangles in 2D star are enumerated by circling around the vertex) need to work with potentials (and not with rotated gradients) We focus for simplicity on FE stars around interior vertices

15 Some notation T a T h : FE star (elements sharing vertex a V h ); domain ω a F a = F s a F b a : faces of the elements in the star T a (2D) skeletal faces F s a (2D) boundary faces F b a Broken H 1 - and H(div)-spaces (broken gradient T ) H 1 (T a) := {v L 2 (ω a) v K H 1 (K) K T a} H(div, T a) := {v L 2 (ω a) v K H(div, K) K T a} Broken polynomial subspaces P p (T a) := {v L 2 (ω a) v K P p (K) K T a} RT p (T a) := {v L 2 (ω a) v K RT p K) K T a}

16 Some notation (cont d) In 3D, a FE star ω a is homeomorphic to a ball in R 3 We can look at the star boundary ω a the traces of the tetrahedra in Ta form a triangulation of ω a every triangle is a boundary face F F b a every edge is the trace of a skeletal face F F s a every point is the trace of a skeletal edge e E a Orientation every skeletal face is oriented so as to define a jump across it every skeletal edge is oriented so as to circle around it incidence coefficients ιf,e = ±1, for all F F e and e E a

17 H 1 -stable polynomial lifting Let p 1 Let r F P p (F ) for all F F a Assume r F 0 on boundary faces F Fa b, and assume on skeletal faces F Fa s the compatibility conditions r F F ωa = 0, ι F,e r F e = 0 e E a F F e Then, min v h P p (T a) v h =r F F F b a [v h ]=r F F F s a T v h ωa C st min v H 1 (T a) v=r F F F b a [v ]=r F F F s a T v ωa with p-robust constant C st only depending on mesh regularity

18 H(div)-stable polynomial lifting Let p 0 Let r K P p (K) for all K T a and let r F P p (F ) for all F F a Assume the compatibility condition (r K, 1) K (r F, 1) F = 0 K T a F F a Then, min v h RT p (T a) v h n F =r F F F b a [v h n F ]=r F F F s a T v h K =r K K T a v h ωa C st min v H(div,T a) v n F =r F F F b a [v n F ]=r F F F s a T v K =r K K T a v ωa with p-robust constant C st only depending on mesh regularity

19 Shifted reformulation: potential Let ξ a h Pp (T a ) be any function from the minimization set ξ a h = r F F F b a, [[ξ a h]] = r F F F s a An equivalent reformulation of Thm. 1 is min T (ξ a v h P p (T a) H0 1(ωa) h v h ) ωa C st min T (ξ a v H0 1(ωa) h v) ωa Application to a posteriori error analysis: ξ a h = ψ au h and r F = 0 F F b a, r F = ψ a [[u h ]] F F s a The compatibility conditions F F e ι F,e r F e = 0, e E a, follow from algebraic properties of jump operator

20 Shifted reformulation: flux Let τ a h RTp (T a ) be any function s.t. τ a h n F = r F F F b a, [[τ a h ]] n F = r F F F s a An equivalent reformulation of Thm. 2 is min τ v h RT p h a +v h ωa C st min (T a) H 0(div,ω a) v h K =r K τ a h K K T a v H 0(div,ω a) v K =r K τ a h K Application to a posteriori error analysis: τ a h = ψ a T u h and r F = 0 F Fa b, r F = ψ a [[ T u h ]] n F F Fa s r K = ψ a (f + T u h ) K T a (f pcw. polynomial) K T a τ a h +v ωa The compatibility condition K T a (r K, 1) K F F a (r F, 1) F = 0 is nothing but Galerkin s orthogonality on the hat basis function ψ a

21 Main ingredients of proofs On a fixed tetrahedron K T a, we can Lift the prescribed divergence of the flux using [Costabel, McIntosh 10] (valid in any space dimension) Lift prescribed polynomials for the flux normal component using [Demkowicz, Gopalakrishnan, Schöberl 12] (proved for d = 3) (a compatibility condition is required if the prescription is on all faces) Lift prescribed compatible polynomials for the potential trace using [DGS 09] (proved for d = 3) We are left with the p-robust lifting of the prescribed jumps... but this requires a careful enumeration of the tetrahedra in the star

22 2D enumeration Circle around the interior vertex a K1: do nothing Kn: fix jump on face touching K n 1, n {2...4} K5: fix last two jumps (possible owing to compatibility condition)

23 3D enumeration: shellability Let T a be a star of tetrahedra around the interior vertex a Consider the triangulation of the sphere S (2) R 3 with same connectivity Enumerate surface triangles so that j i T (2) j is connected for all i The notion of shellability of polytopes shows that this is possible [Ziegler, Lectures on Polytopes, Chap. 8, Springer, 2006] Enumerate patch tetrahedra following surface triangle enumeration and, for each K i T a, fix jump on the skeletal faces of K i touching any tetrahedron K j for j < i

24 Numerical results (with V. Dolejší) Smooth analytical solution in Ω = (0, 1) 2 u(x1, x 2) = sin(2πx 1) sin(2πx 2) Uniformly refined, non-nested unstructured triangulations Discretization by symmetric IPDG method asymptotic exactness observed for pol. degrees p {1...6} similar results for incomplete version of IPDG slightly larger effectivity indices for nonsymmetric version and even p

25 Errors, estimators, effectivity indices h/h 0 p e j(e) e + j(e) η F ηosc η NC η η + j(u h ) I eff I j eff E E E E E E E E / E E E E E E E E / E E E E E E E E / E E E E E E E E E E E E E E E E / E E E E E E E E / E E E E E E E E / E E E E E E E E E E E E E E E E / E E E E E E E E / E E E E E E E E / E E E E E E E E E E E E E E E E / E E E E E E E E / E E E E E E E E / E E E E E E E E E E E E E E E E / E E E E E E E E / E E E E E E E E / E E E E E E E E E E E E E E E E / E E E E E E E E / E E E E E E E E

26 hp-adaptivity Nested simplicial meshes allowing for hanging nodes Extension of reconstruction procedures to hanging nodes only matching refinement of individual patches is needed Bulk chasing criterion based on local p-robust estimators hp-refinement decision criteria inspired from [Mitchell, McClain 14] Algebraic convergence w.r.t. dof s observed on several 2D benchmark problems from [Mitchell 13]

27 Numerical examples Re-entrant corner singularity (u H 1+t (Ω), t < 2 3 ) error in energy norm 1e-01 1e-02 1e-03 1e-04 ideal param, γ =0.30 param, γ =0.60 prior hp decay h-adapt 1e Ε Ε Ε Ε Ε Ε 02 DoF 1/3 Multiple difficulties (re-entrant corner, point sing, circular wave) error in energy norm param, γ =0.30 param, γ =0.60 prior hp decay DoF 1/3 1.0Ε Ε Ε+00

28 Global H(div)-liftings (with I. Smears) So far, we devised local liftings of polynomial data on FE stars We now consider global H(div)-liftings of polynomial data One important application is to devise liftings on patches of mesh cells that are (much) larger than a star This allows us to remove some theoretical barriers on number of hanging nodes for elliptic problems level of mesh coarsening between time-steps in parabolic problems T n 1 \ T n a a T n T a,n ωa ω a

29 Main result Lipschitz domain Ω R d ; boundary partition Γ = Γ D Γ N Let T h be a simplicial mesh of Ω, possibly locally refined Thm. Let p 1, f P p 1 (T h ), ξ RT p 1 (T h ) (broken spaces) (with (f, 1) Ω = 0 if Γ N = Γ). Then, min v h RT p (T h ) H(div,Ω) v h =f in Ω v h n=0 on Γ N ξ + v h Ω C st min v H(div,Ω) v=f in Ω v n=0 on Γ N ξ + v Ω with p-robust constant C st only depending on mesh regularity (Note that the discrete minimizer is one polynomial order higher than the data) See [Ainsworth, Guzman, Sayas 16] for zero interior source terms, nonzero boundary traces, and fixed polynomial degree

30 Main idea in proof Primal problem with H 1 data u H (Ω) 1 s.t. ( u, v) Ω = (f, v) Ω (ξ, v) Ω, v H (Ω) 1 H (Ω) 1 = {v H 1 (Ω) (v, 1) Ω = 0} if Γ N = Γ H 1(Ω) = {v H1 (Ω) v ΓD = 0} otherwise Equivalence of primal/dual energies (f, v) Ω (ξ, v) Ω min ξ + v Ω = max = u Ω v H(div,Ω) v H 1 (Ω) v Ω v=f in Ω v n=0 on Γ N Need to construct σ h RT p (T h ) H(div, Ω) s.t. ξ + σ h Ω C st u Ω

31 A posteriori error estimate with H 1 data Consider general data f L 2 (Ω) and ξ L 2 (Ω) u H (Ω) 1 s.t. ( u, v) Ω = (f, v) Ω (ξ, v) Ω, v H (Ω) 1 H -conforming 1 approximation u h P p (T h ) H (Ω), 1 p 1 Equilibrated flux reconstruction σ h RT p (T h ) H(div, Ω), p p (u u h ) 2 Ω ( uh + ξ + σ h K + osc(f, K) ) 2 K T h u h + ξ + σ h K C eff (u u h ) ωk + osc(f, ξ, ω K ) σ h constructed locally from local shifted flux equilibration in V a h = RTp (T a ) H(div, ω a ) (+ Neumann BC s) σ a h := arg min τ v h Vh a, v h a + v h ωa h=gh a with data τ a h := ψ a(ξ + u h ), g a h := Π ( V a h ) (f ψ a (ξ + u h ) ψ a )

32 Conclusion of proof Consider polynomial data f P p 1 (T h ), ξ RT p 1 (T h ), p 1 Consider H 1 -conforming FEM approximation with 1 = p p The data oscillation term osc(f, ξ, ω K ) vanishes so that u h + ξ + σ h Ω C eff (u u h ) Ω Combined with the basic bound u h Ω u Ω, we conclude that ξ + σ h Ω u h + ξ + σ h Ω + u h Ω (2C eff + 1) u Ω

33 Sharper stability result for special data Let ψ be continuous, pcw. affine on T h ψ can be viewed as a large-scale hat basis function Let Γ N be the subset of Γ where ψ vanishes Let p 1, f P p 1 (T h ), ξ RT p 1 (T h ) (with (f, 1) Ω = 0 if Γ N = Γ) Thm. The following holds: min v h RT p (T h ) H(div,Ω) v h =ψ f ψ ξ in Ω v h n=0 on Γ N ξ + v h Ω C st C(ψ ) min v H(div,Ω) v=f in Ω no BC ξ + v Ω with p-robust constant C st only depending on mesh regularity and C(ψ ) = ψ L (Ω) + h Ω ψ L (Ω) lifting and data have matching polynomial order the function ψ is factored out from RHS

34 Conclusions Equilibrated-flux estimates offer several benefits guaranteed (fully computable) upper bounds p-robust local efficiency adaptive inexact Newton solvers [AE, MV 13] Unified analysis for p-robust H 1 - and H(div)-polynomial liftings New local efficiency proofs for arbitrary-level of hanging nodes (elliptic PDEs) and no coarsening restriction (parabolic PDEs) Thank you for your attention 1. AE, MV, SINUM (2015), 53, V. Dolejší, AE, MV, SISC (2016), 38 A3220 A AE, MV (2016), arxiv AE, I. Smears, MV, Calcolo (2017); DOI /s