Unified A Posteriori Error Control for all Nonstandard Finite Elements 1

Transcription

1 Unified A Posteriori Error Control for all Nonstandard Finite Elements 1 Martin Eigel C. Carstensen, C. Löbhard, R.H.W. Hoppe Humboldt-Universität zu Berlin we know of Guidelines for Applicants M. Eigel (Humboldt) Unified ErrorThese Analysis guidelines should help you EFEF, as you Warwick think about 2010 applying 1 / to 22the

2 Model Example Flux or stress field p in equilibrium equation g + div p = 0 is approximated by piecewise constant p l and yields equilibrium residual Res(v) := (g v p l : D l v) dx for v V := H0 1 (Ω; R m ) Ω Endow V with norm v V := Dv L 2 (Ω) s.t. V H 1 (Ω) and Res(v) div(p p l ) V = Res V := sup v 0 v V Estimation by edge residuals η E := E (p l T+ ν T+ + p l T ν T ) on each interior edge E = T + T with outer unit normals ν T± ( ) 1/2 p l η E := η E 2 T+ E E ν T+ E ν T p l T M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

3 Estimation of Equilibrium Error Up to data oscillation osc(g, ), edge estimator η E is reliable & efficient Res V η E ± osc(g, elements edges) Remark: Edge residual estimator is equivalent to many others (cf. Ainsworth/Oden, Babuška/Strouboulis, Verfürth) Example: For triangulation of Ω R 2 and first-order conforming or nonconforming FEM p l := D l u l and V c l := P 1(T l ; R m ) V ker Res M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

4 Schemes & Applications CR Wilson Han NR (M) NR (A) CNR DSSY Laplace CR KS Han NR (M) NR (A) HMS CJY Stokes BS KS Zhang Ming LLS HMS Elasticity cf. C. Carstensen, Hu Jun, A. Orlando M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

5 Unified Analysis of... applications: Laplace, Stokes, Navier-Lamé, Maxwell equations... schemes: (all?!) conforming, nonconforming, tri/quad, mixed, mortar elements, dg, hanging nodes... M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

6 Goals of Unified Analysis Generalise analysis to cover many different discretisation schemes and applications in one framework Reduce repetition of similar mathematical arguments and focus on specific properties/difficulties No optimal constants but common point of departure and guiding principles M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

7 A Unified Approach Generic Approach For each Application: Verify error residual For each Scheme: Determine discrete space V l ker(residual) and design computable lower/upper bounds of residual Topics Mixed Setting for Unified A Posteriori Error Control Unified Equilibrium Estimator Analysis of Consistency Residual Applications: Poisson, Lamé, Stokes,... M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

8 Mixed Setting for Unified Analysis Abstract problem formulation: Given X l Y l X Y, A : X Y (X Y ) l := l X + l Y (X Y ), find (x, y) X Y s.t. (PM) A(x, y)(ξ, η) = l(ξ, η) for all (ξ, η) X Y Given a (X X ), Λ : Y X, b (X Y ) with b(, v) := a(λv, ) c (Y Y ), A(x, y) := a(x, ) b(, y) + b(x, ) + c(y, ) (PM) then reads a(x, ξ) b(ξ, y) = l X (ξ) b(x, η) + c(y, η) = l Y (η) for all ξ X for all η Y M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

9 Mixed Setting for Unified Analysis Abstract problem formulation: Given X l Y l X Y, A : X Y (X Y ) l := l X + l Y (X Y ), find (x, y) X Y s.t. (PM) A(x, y)(ξ, η) = l(ξ, η) for all (ξ, η) X Y Given a (X X ), Λ : Y X, b (X Y ) with b(, v) := a(λv, ) c (Y Y ), A(x, y) := a(x, ) b(, y) + b(x, ) + c(y, ) (PM) then reads a(x, ξ) b(ξ, y) = l X (ξ) b(x, η) + c(y, η) = l Y (η) for all ξ X for all η Y M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

10 Errors & Residuals in Unified Analysis Given approx. (x l, ỹ l ) X l Y to (x, y), define Res := Res X + Res Y (consistency) (equilibrium) Res X := l X a(x l, ) + b(, ỹ l ) = l X a(x l Λỹ l ) Res Y := l Y b(x l, ) c(ỹ l, ) Remarks: ỹ l Y close to y l, not necessarily discrete Res X involves piecewise gradient D l (y l ỹ l ) of y l ỹ l / Y Since A isomorphism, error residual, i.e. x x l X + y ỹ l Y Res X X + Res Y Y M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

11 Errors & Residuals in Unified Analysis Given approx. (x l, ỹ l ) X l Y to (x, y), define Res := Res X + Res Y (consistency) (equilibrium) Res X := l X a(x l, ) + b(, ỹ l ) = l X a(x l Λỹ l ) Res Y := l Y b(x l, ) c(ỹ l, ) Remarks: ỹ l Y close to y l, not necessarily discrete Res X involves piecewise gradient D l (y l ỹ l ) of y l ỹ l / Y Since A isomorphism, error residual, i.e. x x l X + y ỹ l Y Res X X + Res Y Y M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

12 [Preparation] Generic Equilibrium Error Analysis Two discrete spaces V c l V V and V nc l H 1 (T l ; R m ) with J V c l Π V nc l (H1) H 1 -stable Clément-type operator J : V Vl c into (conforming) subspace Vl c V with first-order approximation property (H2) Vl c and V nc l piecewise smooth w.r.t. shape-regular T l (H3) For p l L 2 (Ω; R m n ) Π : Vl c V l nc s.t. v l Vl c T T l D(Πv l ) L 2 (T ) Dv l L 2 (ω T ) v l dx = Πv l dx T T p l : D l v l dx = p l : D l (Πv l ) dx Ω Ω M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

13 [Result] Generic Equilibrium Error Analysis Thm.[CEHL10 + ]: Suppose R T L 2 (T l ), R E L 2 ( E l ) and Res : Vl nc + V R reads Res(v) := R T v dx + S R E v ds El Suppose (H1)-(H3) and Vl nc ker Res Then η l := h E R E 2 L 2 (E) E El is reliable and efficient in the sense Ω 1/2 = h 1/2 E R E L 2 ( S E l ) η l osc(r T, T l ) osc(r E, E l ) Res V η l + osc(r T, {ω z : z K l }) M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

14 [Result] Generic Equilibrium Error Analysis Thm.[CEHL10 + ]: Suppose R T L 2 (T l ), R E L 2 ( E l ) and Res : Vl nc + V R reads Res(v) := R T v dx + S R E v ds El Suppose (H1)-(H3) and Vl nc ker Res Then η l := h E R E 2 L 2 (E) E El is reliable and efficient in the sense Ω 1/2 = h 1/2 E R E L 2 ( S E l ) η l osc(r T, T l ) osc(r E, E l ) Res V η l + osc(r T, {ω z : z K l }) M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

15 Analysis of Consistency Error Thm.[CEHL10 + ]: Given X = L 2 (Ω), x l = D l y l X l = P 0 (T l ; R n ), Y = H 1 0 (Ω), y l Y l = CR 1 (T l ) and µ l := min η Y x l Dη L 2 (Ω) µ l min η l P 1 (T l ) Y x l D l η l L 2 (Ω) min η l P 1 (T l ) Y h 1 T (y l η l ) L 2 (Ω) h 1 T (y l Sy l ) L 2 (Ω) h 1/2 E [y l ] L 2 ( S E l ) h 1/2 E [D ly l τ E ] L 2 ( S E l ) M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

16 Analysis of Consistency Error Thm.[CEHL10 + ]: Given X = L 2 (Ω), x l = D l y l X l = P 0 (T l ; R n ), Y = H 1 0 (Ω), y l Y l = CR 1 (T l ) and µ l := min η Y x l Dη L 2 (Ω) µ l min η l P 1 (T l ) Y x l D l η l L 2 (Ω) min η l P 1 (T l ) Y h 1 T (y l η l ) L 2 (Ω) h 1 T (y l Sy l ) L 2 (Ω) h 1/2 E [y l ] L 2 ( S E l ) h 1/2 E [D ly l τ E ] L 2 ( S E l ) M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

17 [Application] Laplace Equation Poisson model problem: u = g in Ω and u = 0 on Ω leads to primal mixed formulation with a(p, q) := p q dx, Λu := D l u, b(q, u) = Ω Ω D l u q dx Given g L 2 (Ω), find (p, u) X Y := L 2 (Ω; R n ) H0 1 (Ω) s.t. A(p, u)(q, v) = (g, v) L 2 (Ω) for all (q, v) X Y Since A isomorphism p p l L 2 + u ũ l H 1 Res X X + Res Y Y M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

18 [Application (cont.)] Laplace Equation Treatment of different schemes in the unified framework: Conforming FEM Discrete solution u l = ũ l Y defines discrete flux p l := Λu l = Du l. Consistency error vanishes, equilibrium residual treated as suggested previously. Nonconforming FEM Discrete solution u l defines discrete flux p l := D l u l. Same analysis for equilibrium residual while consistency residual p l Dũ l L 2 (Ω) can be bounded as before. Mixed FEM Discrete solution is discrete flux p l and u l / Y is Lagrange multiplier. Equilibrium residual reduces to osc(g, elements), consistency residual minũl H 1 0 (Ω) p l Dũ l L 2 (Ω) can be bounded as in [Carstensen, Math. Comp. 1997]. M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

19 [Application] Stokes Equations Let a(u, v) := Ω uv dx, Λu := div u, c(u, v) := Ω 2µε(u) : ε(v) dx l X (p, q) := Ω pq dx and X := L2 0 (Ω; Rn ), Y := H 1 0 (Ω; Rn ) With ε(v) := symdv and deviatoric operator dev σ := σ 1 n (tr(σ))1 the linear operator A : X Y (X Y ) reads A(σ, u)(τ, v) := 1 2µ (dev σ, dev τ) L 2 (Ω) (σ, ε(v)) L 2 (Ω) (τ, ε(u)) L 2 (Ω) A isomorphism... M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

20 [Application (cont.)] Stokes Equations Stokes problem: Given g L 2 (Ω; R n ), find (σ, u) X Y s.t. A(σ, u)(τ, v) = (g, v) L 2 (Ω) for all (τ, v) X Y Conforming or nonconforming FEM yield u l and p l with σ = 2µε(u) p1 and σ l = 2µε l (u l ) p l 1 Error control: Given FE solution (σ l, u l ) to (σ, u), then σ σ l L 2 (Ω) min ũ l Y 2µ ε l(u l ũ l ) L 2 (Ω) + Res Y Y + div l u l L 2 (Ω) Examples: all known conforming and nonconforming FEM M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

21 [Application] Linear Elasticity Fourth-order elasticity tensor for λ, µ > 0, Cτ := λ tr(τ)1 + 2µτ for all τ R n n Let a(σ, τ) := Ω (C 1 σ) : τ dx, X := L 2 (Ω; R n n sym ), Y := H 1 0 (Ω; Rn ) Λu := Cε(u) and The linear operator A : X Y (X Y ) then reads A(σ, u)(τ, v) := (C 1 σ, τ) L 2 (Ω) (σ, ε(v)) L 2 (Ω) (τ, ε(u)) L 2 (Ω) A isomorphism, λ-independent operator norms of A and A 1 M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

22 [Application (cont.)] Linear Elasticity Navier-Lamé problem: Given g L 2 (Ω; R n ), find (σ, u) X Y s.t. A(σ, u)(τ, v) = (g, v) L 2 (Ω) for (τ, v) X Y For conforming or nonconforming FEM, σ = Cε(u) and σ l = Cε l (u l ) Error control: Given FE solution (σ l, u l ) to (σ, u), is robust for λ σ σ l L 2 (Ω) min ũ l Y ε l(u l ũ l ) L 2 (Ω) + Res Y Y Examples: Conforming FEM, Kouhia&Stenberg, PEERS M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

23 Conclusions for Unified Analysis Framework for unified a posteriori analysis covers large class of applications & discretisation schemes Similarities are exposed and encountered obstacles/challenges guide development of specific error estimators Approach doesn t strive for most accurate/efficient estimators but rather provides an initial line of attack for as many problems as possible M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

24 Wrap-Up of Unified A Posteriori Analysis C. Carstensen: A unifying theory of a posteriori finite element error control (Numer. Math. 2005) C. Carstensen, Jun Hu, A. Orlando: Framework for the a posteriori error analysis of nonconforming finite elements (SINUM 2007) C. Carstensen, Jun Hu: A Unifying Theory of A Posteriori Error Control for Nonconforming FEM (Numer. Math. 2007) C. Carstensen, ME, R.H.W. Hoppe, C. Löbhard: A Unifying Theory of A Posteriori Error Control ( ) M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

25 Extensions of Unified Analysis Unified Analysis has also been applied/extended to Maxwell Equations [C. Carstensen, R.H.W. Hoppe 2009] Mortar FEM and higher order FEM dg FEM [C. Carstensen, T. Gudi, M. Jensen 2009] Hanging nodes [C. Carstensen, Jun Hu 2008] M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22

26 References C. Carstensen, ME, R.H.W. Hoppe, C. Löbhard: Unified A Posteriori Error Control, in preparation C. Carstensen: A unifying theory of a posteriori finite element error control, Numer. Math. 100 (2005) C. Carstensen, Jun Hu, A. Orlando: Framework for the a posteriori error analysis of nonconforming finite elements, SINUM 45, 1 (2007) C. Carstensen, Jun Hu: A Unifying Theory of A Posteriori Error Control for Nonconforming Finite Element Methods, Numer. Math. (2007), DOI /s z C. Carstensen, T. Gudi, M. Jensen: A unifying theory of a posteriori error control for discontinuous Galerkin FEM C. Carstensen, R.H.W. Hoppe: Unified framework for an a posteriori analysis of non-standard finite element approximations of H(curl)-elliptic problems M. Eigel (Humboldt) Unified Error Analysis EFEF, Warwick / 22