A Posteriori Error Estimation Techniques for Finite Element Methods. Zhiqiang Cai Purdue University

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1 A Posteriori Error Estimation Techniques for Finite Element Methods Zhiqiang Cai Purdue University Department of Mathematics, Purdue University Slide 1, March 16, 2017

2 Books Ainsworth & Oden, A posteriori error estimation in finite element analysis, John Wiley & Sons, Inc., Babǔska & Strouboulis, Finite element method and its reliability, Oxford Science Publication, New York, Demkowicz, Computing with hp-adaptive finite elements, Chapman & Hall/CRC, New York, Verfürth, A review of a posteriori error estimation and adaptive mesh refinement techniques, John Wiley, Verfürth, A posteriori error estimation techniques for finite element methods, Oxford University Press, Oberwolfach Workshop September 4-9, 2016 Self-adaptive numerical methods for comput. challenging problems Department of Mathematics, Purdue University Slide 2, March 16, 2017

3 Outline Introduction Explicit Residual Estimators Zienkiewicz-Zhu (ZZ) Estimators Estimators through Duality Estimators through Least-Squares Principle Acknowledgement This work is supported in part by the National Science Foundation and by Lawrence Livermore National Laboratory. Department of Mathematics, Purdue University Slide 3, March 16, 2017

4 Motivations of A Posteriori Error Estimation Error Control or Solution Verification Adaptive Control of Numerical Algorithms Department of Mathematics, Purdue University Slide 4, March 16, 2017

5 Error Control in Numerical PDEs continuous problem L u = f in Ω find u V s.t. a(u, v) = f(v) v V discrete problem L T u T = f T find u T V T s.t. a(u T, v) = f(v) v V T error control or solution verification Given a tolerance ɛ u u T ɛ Department of Mathematics, Purdue University Slide 5, March 16, 2017

6 A Priori Error Estimation a priori error estimation (convergence) u u T C(u) h α T 0 as h T 0 error control Given a tolerance ɛ ( ) 1/α ɛ C(u) h α ɛ = h = u u T T C(u) T ɛ how to get the a priori error estimation? discrete stability + consistency (smoothness) = convergence Department of Mathematics, Purdue University Slide 6, March 16, 2017

7 A Posteriori Error Estimation a posteriori error estimation compute a quantity η(u T ), estimator, such that u u T C r η(u T ) (reliability bound) where C r is a constant independent of the solution error control Given a tolerance ɛ for known C r, η(u T ) ɛ/c r = u u T ɛ how to get the reliability bound? stability Department of Mathematics, Purdue University Slide 7, March 16, 2017

8 Adaptive Control of Numerical Algorithms If the current approximation u T is not good enough adaptive global mesh or degree refinement adaptive local mesh or degree refinement compute quantities η K (u T ), indicators, for all K T such that η K C e u u T ωk (efficiency bound) Department of Mathematics, Purdue University Slide 8, March 16, 2017

9 Adaptive Mesh Refinement (AMR) Algorithm AMR algorithm Given the data of the underlying PDEs and a tolerance ɛ, compute a numerical soltuion with an error less than ɛ. (1) Construct an initial coarse mesh T 0 representing sufficiently well the geometry and the data of the problem. Set k = 0. (2) Solve the discrete problem on T k. (3) For each element K T k, compute an error indicator η K. (4) If the global estimate η is less than ɛ, then stop. Otherwise, locally refine the mesh T k to construct the next mesh T k+1. Replace k by k + 1 and return to Step (2). Department of Mathematics, Purdue University Slide 9, March 16, 2017

10 Adaptive Mesh Refinement (AMR) Algorithm AMR algorithm Solve Estimate Mark Refine Department of Mathematics, Purdue University Slide 10, March 16, 2017

11 A Posteriori Error Estimation computation of the a posteriori error estimation indicator η K a computable quantity for each K T estimator η = ( K T η 2 K ) 1/2 theory of the a posteriori error estimation reliability bound for error control u u T C r η efficiency bound for efficiency of AMR algorithms η K C e u u T ωk K T Department of Mathematics, Purdue University Slide 11, March 16, 2017

12 Construction of Error Estimators a difficult task u =? e = u u T =? impossible ( 1/2 e = e K) 2 =? doable K T possible avenues residual estimator r = Le = L(u u T ) = f Lu T Zienkiewicz-Zhu (ZZ) estimator ρ T u T duality estimator α 1/2 (σ T + α u T ) with σ T = f Department of Mathematics, Purdue University Slide 12, March 16, 2017

13 elliptic interface problems (α(x) u) = f Interface Problems in Ω R d u = g on Γ D and n (α(x) u) = h on Γ N where α(x) is positive piecewise constant w.r.t Ω = n i=1 Ω i : α(x) = α i > 0 in Ω i smoothness u H 1+β (Ω) where β > 0 could be very small. Department of Mathematics, Purdue University Slide 13, March 16, 2017

14 A Test Problem with Intersecting Interfaces the Kellogg test problem Ω = ( 1, 1) 2, Γ D = Ω, f = 0 and α(x) = in (0, 1) 2 ( 1, 0) 2 1 in Ω \ ([0, 1] 2 [ 1, 0] 2 ) exact solution with µ(θ) being smooth u(r, θ) = r 0.1 µ(θ) H 1.1 ɛ (Ω) Department of Mathematics, Purdue University Slide 14, March 16, 2017

15 Explicit Residual Estimator conforming finite element approximation find u T V T V = HD 1 (Ω) such that a(u T, v) = f(v) v V T residual functional r(v) = f(v) a(u T, v) = a(u u T, v) v V = a(u u T, v v I ) = α (u u T ) (v v I ) dx K T K = + div (α u T ))(v v I ) dx + K T K(f [n e (α u T )] e (v v I ) ds e E e Department of Mathematics, Purdue University Slide 15, March 16, 2017

16 Explicit Residual Estimator L 2 representation of the residual functional r(v) = + div (α u T ))(v v I ) dx + K T K(f [n e (α u T )] e (v v I ) ds e E e global reliability bound u u T C sup 0 v (V, ) r(v) v C ( K T η 2 K ) 1/2 examples of the explicit residual indicator Babuska & Rheinboldt 79 (1D), Babuska & Miller 87 (2D) η 2 K = h 2 K f + div (α u T ) 2 K h e [n e (α u T )] 2 e e K Department of Mathematics, Purdue University Slide 16, March 16, 2017

17 interface test problem exact solution u(r, θ) = r 0.1 µ(θ) H 1.1 ɛ (Ω) Department of Mathematics, Purdue University Slide 17, March 16, 2017

18 global reliability bound Explicit Residual Estimator u u T C sup 0 v (V, ) r(v) v C ( K T η 2 K ) 1/2 examples of the explicit residual indicator Babuska & Rheinboldt 79 (1D), Babuska & Miller 87 (2D) η 2 K = h 2 K f + div (α u T ) 2 K h e [n e (α u T )] 2 e e K Bernardi & Verfürth 2000, Petzoldt 2002 ηk 2 = h 2 α 1 2 K (f + div (α ut )) h K e α 1 2 e 2 e K [n e (α u T )] 2 e Department of Mathematics, Purdue University Slide 18, March 16, 2017

19 meshes generated by BM and BV indicators Department of Mathematics, Purdue University Slide 19, March 16, 2017

20 Robust Efficiency and Reliability Bounds (Bernardi & Verfürth 2000, Petzoldt 2002) robust efficiency bound η K C α 1/2 (u u T ) 0,ωK where C is independent of the jump of α Quasi-Monotonicity Assumption (QMA): any two different subdomains Ω i and Ω j, which share at least one point, have a connected path passing from Ω i to Ω j through adjacent subdomains such that the diffusion coefficient α(x) is monotone along this path. robust reliability bound under the QMA, there exists a constant C independent of the jump of α such that α 1/2 (u u T ) 0,Ω C η Department of Mathematics, Purdue University Slide 20, March 16, 2017

21 Robust Estimator for Nonconforming Elements without QMA in both Two and Three Dimensions (C.-He-Zhang, Math Comp 2017 and SINUM 2017) for nonconforming linear elements, let e = u u T η r,k = h K f 0 0,K, η j,n,f = αk and η j,u,f = indicator and estimator η 2 = K T and let h F α F, A j n,f 0,F, αf, H h F [u h ] 0,F with j n,f = [α h u h n] F η 2 K with η2 K = η 2 r,k F E ( η 2 j,n,f + η 2 j,u,f ) L 2 representation of the error in the energy norm α 1/2 h e 2 = (f,e e I ) K j n,f {e e I } w ds {α e n} w [u T ]ds K T F F E F F E I Department of Mathematics, Purdue University Slide 21, March 16, 2017

22 Robust Estimator for Discontinunous Elements without QMA in both Two and Three Dimensions (C.-He-Zhang, SINUM 2017) for discontinuous elements, let e = u u T and let η r,k = h K αk f k 1 + (α u T ) 0,K, indicator and estimator η 2 = K T η 2 K with η2 K = η 2 r,k F E ( η 2 j,n,f + η 2 j,u,f ) L 2 representation of the error in the energy norm α 1/2 h e 2 = (f k 1 + (α u T ), e ē K ) K j n,f {e ē K } w ds K T F E F I {α e n} w [u T ]ds γ α F, H [u F E F F E F h T ] [ē K ]ds F Department of Mathematics, Purdue University Slide 22, March 16, 2017

23 Zienkiewicz-Zhu (ZZ) Error Estimators ZZ estimators ξ G = ρ T u T where ρ T U T C 0 (Ω) d is a recovered gradient recovery operators Zienkiewicz & Zhu estimator (87, cited 2600 times) ρ T (z) = 1 u ω z T dx z N ω z L 2 -projection find ρ T U h such that ρ T u T = min γ U T γ u T other recovery techniques Bank-Xu, Carstensen, Schatz-Wahlbin, Z. Zhang,... Department of Mathematics, Purdue University Slide 23, March 16, 2017

24 Zienkiewicz-Zhu (ZZ) Error Estimators recovery-based estimators ξ ZZ = ρ T u T theory saturation assumption: there exists a constant β [0, 1) s.t. u ρ T ) β u u T = 1 β ρ T ξ ZZ u 1 + β efficiency and reliability bounds Carstensen, Zhou, etc. Department of Mathematics, Purdue University Slide 24, March 16, 2017

25 Zienkiewicz-Zhu (ZZ) Error Estimators Pro and Con + simple universal asymptotically exact inefficiency for nonsmooth problems unreliable on coarse meshes higher-order finite elements, complex systems, etc. Department of Mathematics, Purdue University Slide 25, March 16, 2017

26 3443 nodes mesh generated by η ZZ Department of Mathematics, Purdue University Slide 26, March 16, 2017

27 Why Does It Fail? true gradient and flux for interface problems u / C 0 (Ω) d recovery space ρ T C 0 (Ω) d the reason of the failure approximating discontinuous functions by continuous functions Department of Mathematics, Purdue University Slide 27, March 16, 2017

28 How to Fix It? (C.-Zhang, SINUM (09, 10, 11), C.-He-Zhang, CMAME 17) true gradient and flux u H 1 (Ω) = u H(curl, Ω) σ = α u H(div, Ω) conforming elements u T H 1 (Ω) = u T H(curl, Ω) σ T = α u T / H(div, Ω) = ˆσ T RT 0 or BDM 1 mixed elements gradient u = α 1 σ H(div, Ω) u T L 2 (Ω), σ T H(div, Ω) ρ T = α 1 σ T / H(curl, Ω) = ˆρ T D 1 or N 1 Department of Mathematics, Purdue University Slide 28, March 16, 2017

29 3557 nodes mesh generated by ξ RT with α = 0.1 Department of Mathematics, Purdue University Slide 29, March 16, 2017

30 Summary of Improved ZZ Error Estimators Pro and Con + simple universal asymptotically exact inefficiency for nonsmooth problems unreliable on coarse meshes higher-order finite elements, complex systems, etc. Department of Mathematics, Purdue University Slide 30, March 16, 2017

31 Estimators through Duality early work Hlaváček, Haslinger, Nećas, and Lovišek, Solution of Vatiational Inequality in Mechanics, Springer-Verlag, New York, (Translation of 1982 book.) Ladevéze-Leguillon (83), Demkowicz and Swierczek (85), Oden, Demkowicz, Rachowicz, and Westermann (89) recent work Vejchodsky (04) Braess-Schöberl (08),... Cai-Zhang (12), with Cao, Falgout, He, Starke,... Department of Mathematics, Purdue University Slide 31, March 16, 2017

32 Estimators through Duality (Equilibrated Residual Error Estimator) (Ladevéze-Leguillon 83, Vejchodsky 04, Braess-Schöberl 08) Prager-Synge identity for u, u T HD 1 (Ω) A 1/2 (u T u) 2 + A 1/2 ( u + A 1 τ ) 2 = A 1/2 ( u T + A 1 τ ) 2 for all τ Σ N (f) {τ H N (div; Ω) τ = f} ( (u T u), A u + τ )) = (u T u, (A u) + τ )) = 0 guaranteed reliable estimator σ T = A 1 u T A 1/2 (u u T ) η (τ ) A 1/2 (τ σ T ) τ Σ N (f) Department of Mathematics, Purdue University Slide 32, March 16, 2017

33 Equilibrated Residual Error Estimator explicit/local calculation of numerical flux piecewise polynomial of degree p 1 w.r.t. T Assume that f is explicit calculation of numerical flux for linear element (Braess-Schöberl 08) ˆσ BS Σ N (f) RT 0 solving a vertex patch mixed problem for p-th order element (Braess-Pillwein-Schöberl 09) ˆσ BPS Σ N (f) RT p 1 p-robust efficiency η K (ˆσ BPS ) C e A 1/2 (u u T ) ωk where C e > 0 is a constant independent of p Department of Mathematics, Purdue University Slide 33, March 16, 2017

34 Non-robustness on constant-free estimator for singular-perturbed reaction- or convection-diffusion problems (Verfürth 09, SINUM) for interface problems (C.-Zhang, SINUM 2012) 10 1 A 1/2 ( u u h ) 0 A 1/2 ( u u h ) 0 and number of nodes Figure 1: mesh generated by η BS Figure 2: error and estimator η BS Department of Mathematics, Purdue University Slide 34, March 16, 2017

35 Robust Equilibrated Residual Error Estimator Prager-Synge identity for all τ Σ N (f) A 1/2 (u u T ) 2 + A 1/2 ( u + A 1 τ ) 2 = A 1/2 (τ σ h ) 2 guaranteed reliable estimator A 1/2 (u u T ) η(τ ) A 1/2 (τ σ T ) τ Σ N (f) inf τ Σ N (f) A 1/2 (τ σ T ) discrete equilibrated flux A 1/2 (ˆσ T σ T ) = min τ Σ N (f) RT p 1 A 1/2 (τ σ T ) For improved ZZ flux, the minimization is over RT 0 or BDM 1 Department of Mathematics, Purdue University Slide 35, March 16, 2017

36 Equilibrated Residual Error Estimator (C.-Zhang, SINUM 2012) explicit/local calculation of numerical flux piecewise polynomial of degree p 1 w.r.t. T Assume that f is explicit calculation of numerical flux for linear element ˆσ CZ Σ N (f) RT 0 solving a vertex patch problem for p-th order element ˆσ CZ Σ N (f) RT p 1 α and p-robust efficiency η K (ˆσ CZ ) C e A 1/2 (u u T ) ωk where C e > 0 is a constant independent of α and p Department of Mathematics, Purdue University Slide 36, March 16, 2017

37 Equilibrated Residual Error Estimator 10 1 A 1/2 ( u u h ) 0 and A 1/2 ( u u h ) number of nodes Figure 3: mesh generated by η CZ Figure 4: η CZ error and estimator Department of Mathematics, Purdue University Slide 37, March 16, 2017

38 minimization problem: Estimators through Duality J(u) = min J(v) v HD 1 (Ω) where J(v) = 1 2 (A v, v) f(v) is the energy functional dual problem: J (σ) = max τ J (τ ) Σ N (f) where J (τ ) = 1 2 (A 1 τ, τ ) is the complimentary functional and Σ N (f) {τ H N (div; Ω) τ = f} duality theory: (Ekeland-Temam 76) J(u) = J (σ) and σ = A u Department of Mathematics, Purdue University Slide 38, March 16, 2017

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40 Summery on Estimators through Duality guaranteed reliability bound A 1/2 (u u T ) η(σ CZ ) where η(σ CZ ) = (J(u T ) J (σ CZ )) 1/2 robust local efficiency bound for A = α I η K (σ CZ ) C e α 1/2 (u u T ) 0,ωK Department of Mathematics, Purdue University Slide 39, March 16, 2017

41 Equilibrated Error Estimator for Discontinuous Elements (C.-He-Starke-Zhang) continuous elements let u c H1 T D (Ω) be an approximation A 1/2 (u u c T ) τ inf Σ N (f) A 1/2 (τ σ h ) where σ T = A u c T is the numerical flux discontinuous elements let u T HD 1 (T ) be an approximation A 1/2 h (u u T ) 2 inf τ Σ N (f) A 1/2 (τ σ T ) 2 + inf A 1/2 ( v ρ v HD 1 (Ω) T ) 2 = inf τ Σ N (f) A 1/2 (τ σ T ) 2 + inf A 1/2 (γ ρ γ H T ) D (curl,ω) where σ T = A u T and ρ T = u T are the numerical flux and gradient, respectively. Department of Mathematics, Purdue University Slide 40, March 16, 2017

42 Equilibrated Error Estimator for Non-conforming Elements of Odd Order discontinuous elements let e = u u nc T A 1/2 h e 2 inf τ Σ N (f) A 1/2 (τ σ T ) 2 + = inf τ Σ N (f) A 1/2 (τ σ T ) 2 + inf A 1/2 ( v ρ v HD 1 (Ω) T ) 2 inf A 1/2 (γ ρ γ H T ) D (curl,ω) explicit calculation of an equilibrated flux for odd order ˆσ T Σ N (f) RT p 1 explicit calculation of a gradient ˆρ T H D (curl, Ω) NE p 1 Department of Mathematics, Purdue University Slide 41, March 16, 2017

43 Equilibrated Error Estimator for Non-conforming Elements of Odd Order discontinuous elements let e = u u nc T A 1/2 h e 2 inf τ Σ N (f) A 1/2 (τ σ T ) 2 + = inf τ Σ N (f) A 1/2 (τ σ T ) 2 + inf A 1/2 ( v ρ v HD 1 (Ω) T ) 2 inf A 1/2 (γ ρ γ H T ) D (curl,ω) indicator and estimator η 2 = K T η 2 K with η2 K = η 2 σ,k + η 2 ρ,k where η σ,k and η ρ,k are given by η σ,k = A 1/2 (ˆσ T σ T ) 0,K and η ρ,k = A 1/2 (ˆρ T ρ T ) 0,K Department of Mathematics, Purdue University Slide 42, March 16, 2017

44 Equilibrated Error Estimator for Non-conforming Elements indicator and estimator of Odd Order η 2 = K T η 2 K with η2 K = η 2 σ,k + η 2 ρ,k where η σ,k and η ρ,k are given by η σ,k = A 1/2 (ˆσ T σ T ) 0,K and η ρ,k = A 1/2 (ˆρ T ρ T ) 0,K α-robust efficiency without QMA ) η K C e ( A 1/2 (u u c ) T ω K + osc(f, ω K ) where C e > 0 is a constant independent of α α-robust reliability without QMA A 1/2 (u u nc T ) C r (η + osc(f, T )) where C r > 0 is a constant independent of α Department of Mathematics, Purdue University Slide 43, March 16, 2017

45 Kellogg s Problem for Crouzeix-Raviart Element conforming error η σ = Energy error u u h A N 0.5 eta Figure 5: mesh generated by η K Figure 6: error and estimator η Department of Mathematics, Purdue University Slide 44, March 16, 2017

46 Poisson s Equation on L-Shape Domain for Crouzeix-Raviart Element 10 0 Energy error u u h A N 0.5 eta Figure 7: mesh generated by η K Figure 8: error and estimator η Department of Mathematics, Purdue University Slide 45, March 16, 2017

47 Summary estimators explicit residual: improved ZZ: reasonable mesh, not efficient error control ignoring equilibrium equation dual: the method of choice adaptive control of meshing (AMR) some nice problems: many viable estimators challenging problems (layers, oscillations, etc) : a few error control asymptotic meshes: some smooth problems non-asymptotic meshes:??? challenging problems :??? Department of Mathematics, Purdue University Slide 46, March 16, 2017

48 Grand Computational Challenges complex systems multi-scales, multi-physics, etc. computational difficulties oscillations interior/boundary layers interface singularities nonlinearity??? a general and viable approach adaptive method + accurate, robust error estimation Department of Mathematics, Purdue University Slide 48, March 16, 2017

49 Adaptive Control of Numerical Algorithms If error is corrected whenever it is recognized as such, the path to error is the path of truth by Hans Reichenbach, the renowned philosopher of science, in his 1951 treatise, The Rise of Scientific Philosophy Department of Mathematics, Purdue University Slide 49, March 16, 2017