A posteriori error estimation in the FEM

Transcription

1 A posteriori error estimation in the FEM

2 Plan 1. Introduction 2. Goal-oriented error estimates 3. Residual error estimates 3.1 Explicit 3.2 Subdomain error estimate 3.3 Self-equilibrated residuals 3.4 element error estimate 4. Post-processing 5. Interpolation 6. Based on duality error estimates April, 2011 A posteriori error estimation in the FEM 2 / 17

3 Introduction Variational formulation: { u V : B(u, v) = L(v) v V (0.1) FEM formulation: u h V h V B(u h, v h ) = L(v h ) v h V h (0.2) Residuum functional: r(v) = B(u h, v) L(v) 0, v V. r(v) = B(u h u, v) (0.3) April, 2011 A posteriori error estimation in the FEM 3 / 17

4 Wstȩp A norm of a linear transformation: A : X Y Av Y A := sup v X v X In particular: L : X R Lv L := sup, norm in the space of linear functionals v X v X How to measure residuum in FEM? r := sup v V r(v) v V = sup v V B(u h u, v) v V April, 2011 A posteriori error estimation in the FEM 4 / 17

5 Introduction We assume that B(, ) is a symmetric, continuous, coercive bilinear form. Then B defines a scalar product in V B(u, v) = B(v, u) B(u, u) 0, B(u, u) = 0 u = 0 For every scalar product the Cauchy-Schwarz inequality holds: B(u, v) B(u, u) 1/2 B(v, v) 1/2 Def. Energy norm: w E = B(w, w) 1/2 April, 2011 A posteriori error estimation in the FEM 5 / 17

6 Introduction Def. Energy associated with function w V : E(w) := 1 B(w, w) L(w) 2 Thrm. { u V : B(u, v) = L(v) v V { u V : E(u) = min Thrm. u h u 2 E = 2[E(u h) E(u)] Proof: B(u, u) = L(u) E(u) = 1B(u, u) B(u, u) = 1 B(u, u) 2 2 We obtain: 2[E(u h ) E(u)] = B(u h, u h ) + B(u, u) = B(u + u h, u u h ) = B(u+u h, u u h ) 2B(u h, u u h ) = B(u u h, u u h ) = u u h 2 E ) April, 2011 A posteriori error estimation in the FEM 6 / 17

7 Introduction Thrm. u E = sup v V B(u, v) v E (Since B(u, v) u E v E ) Application: r(v) = B(u u h, v) v V. According to the theorem: u u h E = sup v V B(u u h, v) v E = sup v V r(v) v E = r Energy norm of the error = dual norm of the residuum Residual error estimates: based on estimation of the norm of the residuum. April, 2011 A posteriori error estimation in the FEM 7 / 17

8 Goal-oriented error estimates Quantity of interest (q.o.i.) functional F ( ), continuous on V. Examples: F = udx, F = u nds, F = u cdx, etc. ω ω How to estimate F (u) F (u h )? Define a generalized Green s function: B(w, G) = F (w), w V Error estimate: F (u) F (u h ) = F (u u h ) = B(u u h, G) = B(u u h, G G h ) K B K(u u h, G G h ) K B K(u u h, u u h ) 1/2 B K (G G h, G G h ) 1/2 = K u u h E,K G G h E,K April, 2011 A posteriori error estimation in the FEM 8 / 17 ω

9 A model elliptic boundary-value problem a u + bu = f in Ω a u n = g on Γ N u = 0 on Γ D B(u, v) = (a u v + buv)dx, Ω L(u) = fvdx + Ω gvds Γ N u V : B(u, v) = L(v) v V April, 2011 A posteriori error estimation in the FEM 9 / 17

10 Explicit residual error estimate where u h u 2 E C K ( ) h 2 K r 2 dx + h K R 2 ds, K K r = a u h bu h + f, 0 na Γ D, R = g a u h n na Γ N, 1 [ a u 2 h n ] na Γ ij. (0.4) April, 2011 A posteriori error estimation in the FEM 10 / 17

11 Subdomain residual method Partition of unity in Ω: { ψi H 1 (Ω), ψ i 0, i = 1,..., N, N i=1 ψ i(x) = 1, x Ω. Example: ψ(x) = (bi/tri)-linear global FE shape functions. Auxiliary solutions: Ω i = supp ψ i { ũi H 1 (Ω i ), ũ i = u h on Ω i Error estimate: B(ũ i, v) = L(v), v H 1 0(Ω i ). N u u h 2 E C ũ i u h 2 E,Ω i, i=1 April, 2011 A posteriori error estimation in the FEM 11 / 17

12 Element residual method Kernel of interpolation operator: Π hp : V p+1 (K) V p (K): M K = {v V h,p+1 (K) : Π hp v = 0}. (bubble functions) Residual r(v) on element K r(v) = [f + a u h bu h ]v dx+ K 1 (g a u n n)v ds + [ a u n ]v ds. K Γ N 2 Auxiliary solutions φ K M K : B(φ K, v) = r(v), v M K, K\ Ω Error estimate: u h u 2 E C K φ K 2 E,K, April, 2011 A posteriori error estimation in the FEM 12 / 17

13 Self-equilibrated residuals Split residual functional into element contributions: r(u h, v) = B(u h, v) L(v) = (B K (u h, v) L K (v)) = K (B K(u h, v) L K (v) λ K (v)) = r K }{{} K (v). K :=r K (v) λ K (v) = 0 v V, r K (ψk) n = 0, n = 1,..., 8. K (consistency and self-equilibration) Auxiliary solutions Find φ K V (K) = H 1 (K), such that B K (φ K, v) = r K (v), v V (K). Error estimate: u h u E,Ω ( φ K 2 E,K )1/2, no constant C! K Self-eqilibration: Kelly, Ainsworth, Ladeveze,... April, 2011 A posteriori error estimation in the FEM 13 / 17

14 Error estimates via post-processing u u h 2 1,Ω = [(u u h ) 2 + u u h 2 ]dx Ω [(u h u h ) 2 + u h u h 2 ]dx Ω where u h and u h are post-processed u h and u h. Mathematical post-processing: Extraction formulas, Babuška, Miller, (constant coefficients). Higher order accuracy by averaging, Bramble, Schatz, (uniform meshes). L 2 projection of u h onto V h Oden, Brauchli, Rachowicz (uniform meshes), Zienkiewicz, Zhu (patches of non-uniform meshes). Superconvergence at Gaussian points, M. Zlamal Post-processing for fluxes, M.F. Miller, J.R. Whiteman April, 2011 A posteriori error estimation in the FEM 14 / 17

15 Duality Functional of potential energy: E(w) = 1 B(w, w) L(w) 2 Functional of complementary energy: E (s) = 1 [ 1 2 a s s + 1 b (f + s)2 ]ds, s K Ω K = {s L 2 (Ω) : s L 2 (Ω), s n = g, on Γ N } Equivalence of BVP to minimization/maximization: B(u, v) = L(v) v V E(u) = min B (r, s) = L (s) s K 0 E (r) = max Relation of solutions: E(u) = E (r), r = a u April, 2011 A posteriori error estimation in the FEM 15 / 17

16 Duality E (r h ) E (r) = E(u) E(u h ) Therefore: 1 2 h E = E(u h ) E(u) E(u h ) E (r h ) 1 2 h E = E (r) E (r h ) E(u h ) E (r h ) Bilinear and linear forms for the complementary problem: B (r, s) = [ 1 Ω a r s + 1 b r s]dx L 1 (s) = f sdx b Ω April, 2011 A posteriori error estimation in the FEM 16 / 17

17 Interpolation error estimates A priori interpolation error estimate u u I 1,K Ch min(p,r 1) p (r 1) u r,k Take r = 2: u u I 1,K C h p u 2,K Lemat Cea: u u h 2 1,Ω C K ( ) 2 hk u p 2,K K Norm u 2,K is replaced by u h 2,K for p 2 or we use finite differencies, if p = 1. April, 2011 A posteriori error estimation in the FEM 17 / 17