Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation

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1 Adaptive Finite Element Methods Lecture 1: A Posteriori Error Estimation Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA rhn 7th Zürich Summer School, August 01 A Posteriori Error Control and Adaptivity

2 Outline Piecewise Polynomial Interpolation in Sobolev Spaces Model Problem and FEM FEM: A Posteriori Error Analysis Surveys

3 Outline Piecewise Polynomial Interpolation in Sobolev Spaces Model Problem and FEM FEM: A Posteriori Error Analysis Surveys

4 Warm-up: 1d Example Question: given a continuous function u : [0, 1] R, a partition T N = {x n } N n=0 with x 0 = 0, x N = 1, and a pw constant approximation U N of u over T N, what is the best decay rate of u U N L (0,1)? Answer 1: W -Regularity. 1 Let u W (0, 1 1) and T N be quasi-uniform. Then U N (x) = u(x n 1 ) for x n 1 x < x n satisfies U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L (0,1). Answer : W 1 1 -Regularity. Let u W 1 1 (0, 1). If x n is defined by xn x n 1 u (s) ds = 1 N u L 1 (0,1), then U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L 1 (0,1).

5 Warm-up: 1d Example Question: given a continuous function u : [0, 1] R, a partition T N = {x n } N n=0 with x 0 = 0, x N = 1, and a pw constant approximation U N of u over T N, what is the best decay rate of u U N L (0,1)? Answer 1: W -Regularity. 1 Let u W (0, 1 1) and T N be quasi-uniform. Then U N (x) = u(x n 1 ) for x n 1 x < x n satisfies U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L (0,1). Answer : W 1 1 -Regularity. Let u W 1 1 (0, 1). If x n is defined by xn x n 1 u (s) ds = 1 N u L 1 (0,1), then U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L 1 (0,1).

6 Warm-up: 1d Example Question: given a continuous function u : [0, 1] R, a partition T N = {x n } N n=0 with x 0 = 0, x N = 1, and a pw constant approximation U N of u over T N, what is the best decay rate of u U N L (0,1)? Answer 1: W -Regularity. 1 Let u W (0, 1 1) and T N be quasi-uniform. Then U N (x) = u(x n 1 ) for x n 1 x < x n satisfies U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L (0,1). Answer : W 1 1 -Regularity. Let u W 1 1 (0, 1). If x n is defined by xn x n 1 u (s) ds = 1 N u L 1 (0,1), then U n (x) u(x) = u(x n 1 ) u(x) xn 1 x u (s) ds 1 N u L 1 (0,1).

7 Sobolev Number Let ω R d be Lipschitz and bounded, k N, 1 p. The Sobolev number of W k p (ω) is sob(w k p ) := k d p. Remark 1. This number governs the scaling properties of seminorm v W k p (ω): consider ˆx = 1 hx which transforms ω into ω and note ˆv W k p (bω) = h sob(w k p ) v W k p (ω) v W k p (ω). Remark. Let d = 1 and ω = (0, 1). Then W 1 (ω) is the linear (and usual) Sobolev scale of L (ω), but W 1 1 (ω) is in the nonlinear scale of L (ω), i.e. sob(w 1 1 ) = = 0 1 = sob(l ).

8 Sobolev Number Let ω R d be Lipschitz and bounded, k N, 1 p. The Sobolev number of W k p (ω) is sob(w k p ) := k d p. Remark 1. This number governs the scaling properties of seminorm v W k p (ω): consider ˆx = 1 hx which transforms ω into ω and note ˆv W k p (bω) = h sob(w k p ) v W k p (ω) v W k p (ω). Remark. Let d = 1 and ω = (0, 1). Then W 1 (ω) is the linear (and usual) Sobolev scale of L (ω), but W 1 1 (ω) is in the nonlinear scale of L (ω), i.e. sob(w 1 1 ) = = 0 1 = sob(l ).

9 Sobolev Number Let ω R d be Lipschitz and bounded, k N, 1 p. The Sobolev number of W k p (ω) is sob(w k p ) := k d p. Remark 1. This number governs the scaling properties of seminorm v W k p (ω): consider ˆx = 1 hx which transforms ω into ω and note ˆv W k p (bω) = h sob(w k p ) v W k p (ω) v W k p (ω). Remark. Let d = 1 and ω = (0, 1). Then W 1 (ω) is the linear (and usual) Sobolev scale of L (ω), but W 1 1 (ω) is in the nonlinear scale of L (ω), i.e. sob(w 1 1 ) = = 0 1 = sob(l ).

10 Conforming Meshes: The Bisection Method and REFINE Labeling of a sequence of conforming refinements T 0 T 1 T for d = (similar but much more intricate for d > ) Shape regularity: the shape-regularity constant of any T T solely depends on the shape-regularity constant of T 0. Nested spaces: refinement leads to V(T ) V(T ) because T T. Monotonicity of meshsize function h T : if h T T := h T := T 1/d, then h T h T for T T, and reduction property with b 1 bisections h T T b/d h T T T T \ T.

11 Complexity of REFINE Recursive bisection of T 3 (sequence of compatible bisection patches) Naive estimate is NOT valid with Λ 0 independent of refinement level #T #T Λ 0 #M Complexity of REFINE (Binev, Dahmen, DeVore 04 (d = ), and Stevenson 07 (d > )): If T 0 has a suitable labeling, then there exists a constant Λ 0 > 0 only depending on T 0 and d such that for all k 1 k 1 #T k #T 0 Λ 0 #M j. j=0

12 Complexity of REFINE Recursive bisection of T 3 (sequence of compatible bisection patches) Naive estimate is NOT valid with Λ 0 independent of refinement level #T #T Λ 0 #M Complexity of REFINE (Binev, Dahmen, DeVore 04 (d = ), and Stevenson 07 (d > )): If T 0 has a suitable labeling, then there exists a constant Λ 0 > 0 only depending on T 0 and d such that for all k 1 k 1 #T k #T 0 Λ 0 #M j. j=0

13 Complexity of REFINE Recursive bisection of T 3 (sequence of compatible bisection patches) Naive estimate is NOT valid with Λ 0 independent of refinement level #T #T Λ 0 #M Complexity of REFINE (Binev, Dahmen, DeVore 04 (d = ), and Stevenson 07 (d > )): If T 0 has a suitable labeling, then there exists a constant Λ 0 > 0 only depending on T 0 and d such that for all k 1 k 1 #T k #T 0 Λ 0 #M j. j=0

14 Piecewise Polynomial Interpolation Quasi-local error estimate: if 0 t s n + 1 (n 1 polynomial degree) and 1 p, q satisfy sob(w s p ) > sob(w t q), then for all T T D t (v I T v) L q (T ) h sob(w s p ) sob(w t q ) T D s v L p (N T (T )), where N T (T ) is a discrete neighborhood of T and I T is a quasi interpolation operator (Clement or Scott-Zhang). If sob(wp s ) > 0, then v is Hölder continuous, I T can be replaced by the Lagrange interpolation operator, and N T (T ) = T. Quasi-uniform meshes: if 1 s n + 1 and u H s (Ω), then (v I T v) L (Ω) v H s (Ω)(#T ) s 1 d. Optimal error decay: If s = n + 1 (linear Sobolev scale), then (v I T v) L (Ω) v H n+1 (Ω)(#T ) n d.

15 Piecewise Polynomial Interpolation Quasi-local error estimate: if 0 t s n + 1 (n 1 polynomial degree) and 1 p, q satisfy sob(w s p ) > sob(w t q), then for all T T D t (v I T v) L q (T ) h sob(w s p ) sob(w t q ) T D s v L p (N T (T )), where N T (T ) is a discrete neighborhood of T and I T is a quasi interpolation operator (Clement or Scott-Zhang). If sob(wp s ) > 0, then v is Hölder continuous, I T can be replaced by the Lagrange interpolation operator, and N T (T ) = T. Quasi-uniform meshes: if 1 s n + 1 and u H s (Ω), then (v I T v) L (Ω) v H s (Ω)(#T ) s 1 d. Optimal error decay: If s = n + 1 (linear Sobolev scale), then (v I T v) L (Ω) v H n+1 (Ω)(#T ) n d.

16 Adaptive Approximation (Binev, Dahmen, DeVore, Petrushev 0) Question: can one achieve the same decay rate with lower regularity? Let n = 1, d = and note that H (Ω) A 1/ where A 1/ = {v H 1 0 (Ω) : inf v I T v H #T #T 1 (Ω) N 1/ } 0 N Let v W p (Ω; T 0 ) H 1 0 (Ω) with p > 1, and notice that sob(w p ) = p > 1 = 0 = sob(h1 ). Theorem 1. Given any δ > 0, the following algorithm THRESHOLD THRESHOLD(T, δ) while M := {T T : (v I T v) L (T ) > δ} T := REFINE(T, M) end while return(t ) terminates and its output satisfies v I T v H 1 (Ω) δ(#t ) 1/, #T #T 0 δ 1 Ω 1 1/p D v L p (Ω;T 0).

17 Adaptive Approximation (Binev, Dahmen, DeVore, Petrushev 0) Question: can one achieve the same decay rate with lower regularity? Let n = 1, d = and note that H (Ω) A 1/ where A 1/ = {v H 1 0 (Ω) : inf v I T v H #T #T 1 (Ω) N 1/ } 0 N Let v W p (Ω; T 0 ) H 1 0 (Ω) with p > 1, and notice that sob(w p ) = p > 1 = 0 = sob(h1 ). Theorem 1. Given any δ > 0, the following algorithm THRESHOLD THRESHOLD(T, δ) while M := {T T : (v I T v) L (T ) > δ} T := REFINE(T, M) end while return(t ) terminates and its output satisfies v I T v H 1 (Ω) δ(#t ) 1/, #T #T 0 δ 1 Ω 1 1/p D v L p (Ω;T 0).

18 Proof of Theorem 1 1. If r := sob(w p ) sob(h 1 ) = p > 0, E T := (v I T v) L (T ) for T T, then E T h r T D v L p (T ), we deduce that THRESHOLD terminates in finite steps k(δ) 1.. Decompose M := k j=0 M j into the sets P j of market elements T : (j+1) T < j (j+1)/ h T < j/. Elements of P j are disjoint for otherwise they are contained in one another contradicting the definition. Hence (j+1) #P j Ω #P j Ω j Since δ E T (j/)r D v L p (T ) for T P j, we get δ p #P j (j/)rp T P j D v p L p (T ) (j/)rp D v p L p (Ω;T 0), whence #P j δ p (j/)rp D v p L p (Ω;T 0).

19 4. The crossover takes place for j 0 such that j0+1 Ω = δ p j0 rp D v p L p (Ω;T 0) j0 δ 1 D v L p (Ω;T 0) Ω 1/p. 5. Compute #M = j #P j Ω j + δ p D v p L p (Ω;T 0) ( rp/ ) j j j 0 j>j 0 ( δ 1 + δ p δ p 1) Ω 1 1/p D v L }{{} p (Ω;T 0). =δ 1 Apply Theorem about complexity of REFINE to arrive at #T #T 0 #M δ 1 Ω 1 1/p D v L p (Ω;T 0). 6. Upon termination of THRESHOLD, E T δ for all T T, whence v I T v H 1 (Ω) = T T E T δ #T.

20 Remarks on Adaptive Approximation Let v H 1 0 (Ω) W p (Ω; T 0 ), n = 1, d =, p > 1. For N > #T 0 there exists T T such that v I T v H 1 (Ω) Ω 1 1/p D v L p (Ω;T 0)N 1/, #T #T 0 N. Choose δ = Ω 1 1/p D v L p (Ω)N 1 in algorithm THRESHOLD. W p (Ω; T 0 ) A 1/ for d = and p > 1. All geometric singularities for d = (corner and interfaces) satisfy this (Nicaise 94). For arbitrary n 1, d, comparing Sobolev numbers yields n + 1 d p > sob(h1 ) = 1 d p > d n + d. This may give p < 1 and corresponding Besov space Bp,p n+1 (Ω). Proof above works. Regularity theory for elliptic PDE is incomplete for p < 1. Anisotropic elements: Isotropic refinement is not always optimal for d = 3.

21 Remarks on Adaptive Approximation Let v H 1 0 (Ω) W p (Ω; T 0 ), n = 1, d =, p > 1. For N > #T 0 there exists T T such that v I T v H 1 (Ω) Ω 1 1/p D v L p (Ω;T 0)N 1/, #T #T 0 N. Choose δ = Ω 1 1/p D v L p (Ω)N 1 in algorithm THRESHOLD. W p (Ω; T 0 ) A 1/ for d = and p > 1. All geometric singularities for d = (corner and interfaces) satisfy this (Nicaise 94). For arbitrary n 1, d, comparing Sobolev numbers yields n + 1 d p > sob(h1 ) = 1 d p > d n + d. This may give p < 1 and corresponding Besov space Bp,p n+1 (Ω). Proof above works. Regularity theory for elliptic PDE is incomplete for p < 1. Anisotropic elements: Isotropic refinement is not always optimal for d = 3.

22 Outline Piecewise Polynomial Interpolation in Sobolev Spaces Model Problem and FEM FEM: A Posteriori Error Analysis Surveys

23 Model Problem: Basic Assumptions Consider model problem with div(a u) = f in Ω, u Ω = 0, Ω polygonal domain in R d, d ; T 0 is a conforming mesh made of simplices with compatible labeling; A(x) is symmetric and positive definite for all x Ω with eigenvalues λ(x) satisfying A is piecewise Lipschitz in T 0 ; 0 < a min λ i (x) a max, x Ω; f L (Ω) (and in Lecture 3 f H 1 (Ω)); V(T ) space of continuous elements of degree n over a conforming refinement T of T 0 (by bisection). Exact numerical integration.

24 Galerkin Method Function space: V := H 1 0 (Ω). Bilinear form: B : V V R B(v, w) := A v w v, w V. Then solution u of model problem satisfies Ω u V : B(u, v) = f, v v V. Finite element space: If P n (T ) denote polynomials of degree n over T, then V(T ) := {v H 1 0 (Ω) : v T P n (T ) T T }. Galerkin solution: The discrete solution U = U T satisfies U V(T ) : B(U, V ) = f, V V V(T ).

25 Galerkin Method (Continued) Residual: R V = H 1 (Ω) is given by R, v := f, v B(U, v) = B(u U, v) v V. Galerkin Orthogonality: R, V = f, V B(U, V ) V V(T ). Quasi-Best (Céa Lemma): α 1 α coercivity and continuity constants of B α 1 u U V B(u U, u U) = B(u U, u V ) α u U V u V V V V(T ). u U V α inf α u V V. 1 V V(T ) Approximation Class A s : Let 0 < s n/d (n 1) and { ( )} A s := v V : u s := sup N s inf inf v V V N>0 #T #T 0 N V V(T ) T T : #T #T 0 N, inf V V(T ) v V V v s N s.

26 A Priori Error Analysis If u A s, 0 < s n/d, there exists T T with #T #T 0 N and u U V α α 1 u s N s. If n = 1, d =, p > 1, and u V W p (Ω; T 0 ), then THRESHOLD shows that u 1/ D u L p (Ω;T 0) whence (optimal estimate) T T : #T #T 0 N, u U V D u L p (Ω;T 0)N 1/. THRESHOLD needs access to the element interpolation error E T and so to the unknown u. It is thus not practical. The a posteriori error analysis provides a tool to extract this missing information from the residual R. This is discussed next. The a priori analysis is valid for a bilinear for B on a Hilbert space V that is continuous and satisfies a discrete inf-sup condition B(v, w) α 1 v V w V v, w V; α V V sup W V B(V, W ) W V V V(T ).

27 A Priori Error Analysis If u A s, 0 < s n/d, there exists T T with #T #T 0 N and u U V α α 1 u s N s. If n = 1, d =, p > 1, and u V W p (Ω; T 0 ), then THRESHOLD shows that u 1/ D u L p (Ω;T 0) whence (optimal estimate) T T : #T #T 0 N, u U V D u L p (Ω;T 0)N 1/. THRESHOLD needs access to the element interpolation error E T and so to the unknown u. It is thus not practical. The a posteriori error analysis provides a tool to extract this missing information from the residual R. This is discussed next. The a priori analysis is valid for a bilinear for B on a Hilbert space V that is continuous and satisfies a discrete inf-sup condition B(v, w) α 1 v V w V v, w V; α V V sup W V B(V, W ) W V V V(T ).

28 A Priori Error Analysis If u A s, 0 < s n/d, there exists T T with #T #T 0 N and u U V α α 1 u s N s. If n = 1, d =, p > 1, and u V W p (Ω; T 0 ), then THRESHOLD shows that u 1/ D u L p (Ω;T 0) whence (optimal estimate) T T : #T #T 0 N, u U V D u L p (Ω;T 0)N 1/. THRESHOLD needs access to the element interpolation error E T and so to the unknown u. It is thus not practical. The a posteriori error analysis provides a tool to extract this missing information from the residual R. This is discussed next. The a priori analysis is valid for a bilinear for B on a Hilbert space V that is continuous and satisfies a discrete inf-sup condition B(v, w) α 1 v V w V v, w V; α V V sup W V B(V, W ) W V V V(T ).

29 Outline Piecewise Polynomial Interpolation in Sobolev Spaces Model Problem and FEM FEM: A Posteriori Error Analysis Surveys

30 Error-Residual Equation (Babuška-Miller 87) Since R, v = f, v B(U, v) = B(u U, v) for all v V, we deduce u U V 1 α 1 R V α α 1 u U V. Residual representation: elementwise integration by parts yields R, v = f + div(a U) v + [A U] ν v v V T T T }{{} S S S }{{} =r =j where r = r(u), j = j(u) are the interior and jump residuals. Localization: The Courant (hat) basis {φ z } z N (T ) satisfy the partition of unity property z N (T ) φ z = 1. Therefore, for all v V, R, v = R, vφ z = ( ) rvφ z + jvφ z. ω z γ z z N (T ) Galerkin orthogonality: z N (T ) ω z rφ z + γ z jφ z = 0 z N 0 (T )

31 Reliability: Global Upper A Posteriori Bound Exploit Galerkin orthogonality R, v = ( ) r(v α z (v))φ z + j(v α z (v))φ z ω z γ z z N (T ) R and take α z (v) := Use Poincaré inequality in ω z ωz vφz Rωz φz if z is interior and α z (v) = 0 if z Ω. v α z (v) L (ω z) C 0 h z v L (ω z) z N (T ) and a scaled trace lemma, to deduce R, vφz ) (h z rφ 1/ z L (ω z) + h 1/ z jφ 1/ z L (γ z) v L (ω z). Sum over z N (T ) and use z N (T ) v L (ω z) v L (Ω) to get ( R V z N (T ) h z rφ 1/ z L (ω z) + h z jφ 1/ z L (γ z)) 1/.

32 Reliability: Global Upper A Posteriori Bound Exploit Galerkin orthogonality R, v = ( ) r(v α z (v))φ z + j(v α z (v))φ z ω z γ z z N (T ) R and take α z (v) := Use Poincaré inequality in ω z ωz vφz Rωz φz if z is interior and α z (v) = 0 if z Ω. v α z (v) L (ω z) C 0 h z v L (ω z) z N (T ) and a scaled trace lemma, to deduce R, vφz ) (h z rφ 1/ z L (ω z) + h 1/ z jφ 1/ z L (γ z) v L (ω z). Sum over z N (T ) and use z N (T ) v L (ω z) v L (Ω) to get ( R V z N (T ) h z rφ 1/ z L (ω z) + h z jφ 1/ z L (γ z)) 1/.

33 Reliability: Global Upper A Posteriori Bound Exploit Galerkin orthogonality R, v = ( ) r(v α z (v))φ z + j(v α z (v))φ z ω z γ z z N (T ) R and take α z (v) := Use Poincaré inequality in ω z ωz vφz Rωz φz if z is interior and α z (v) = 0 if z Ω. v α z (v) L (ω z) C 0 h z v L (ω z) z N (T ) and a scaled trace lemma, to deduce R, vφz ) (h z rφ 1/ z L (ω z) + h 1/ z jφ 1/ z L (γ z) v L (ω z). Sum over z N (T ) and use z N (T ) v L (ω z) v L (Ω) to get ( R V z N (T ) h z rφ 1/ z L (ω z) + h z jφ 1/ z L (γ z)) 1/.

34 Upper A Posteriori Bound (Continued) Use that h z h(x) for all x ω z, and z N (T ) φ z = 1, to derive ( ) 1/ R V hr L (Ω) + h1/ j L (Γ) in terms of weighted (and computable) L norms of the residuals. Upper bound: Introduce element indicators E T (U, T ) E T (U, T ) = h T r L (T ) + h T j L ( T ) and error estimator E T (U) = T T E T (U, T ). Then u U V 1 R V 1 E T (U). α 1 α 1 Jump residual dominates interior residual: Let r z = r,φz φ z,1 R and note ω z r(v α z (v))φ z = ω z (r r z )(v α z (v))φ z. Then ( R V z N (T ) h z (r r z )φ 1/ z L (ω z) + h z jφ 1/ z L (γ z)) 1/.

35 Upper A Posteriori Bound (Continued) Use that h z h(x) for all x ω z, and z N (T ) φ z = 1, to derive ( ) 1/ R V hr L (Ω) + h1/ j L (Γ) in terms of weighted (and computable) L norms of the residuals. Upper bound: Introduce element indicators E T (U, T ) E T (U, T ) = h T r L (T ) + h T j L ( T ) and error estimator E T (U) = T T E T (U, T ). Then u U V 1 R V 1 E T (U). α 1 α 1 Jump residual dominates interior residual: Let r z = r,φz φ z,1 R and note ω z r(v α z (v))φ z = ω z (r r z )(v α z (v))φ z. Then ( R V z N (T ) h z (r r z )φ 1/ z L (ω z) + h z jφ 1/ z L (γ z)) 1/.

36 Efficiency: Local Lower A Posteriori Bound (n = 1) (Verfürth 89) Local dual norms: for v H 1 0 (ω) we have R, v = B(u U, v) α u U V v V R H 1 (ω) α u U V Interior residual: take ω = T T and note R, v = rv. Then T R H 1 (T ) = r H 1 (T ) Overestimation: Poincaré inequality yields r H 1 (T ) h T r L (T ) rv r L (T ) v L (T ) h T r L (T ) v L (T ) T Pw constant r: Let η H0 1 (T ), T T η, η L (T ) h 1 T. Then r L (T ) r(rη) r H 1 (T ) r L (T ) η L (T ) T h 1 T r H 1 (T ) r L (T ) h T r L (T ) r H 1 (T )

37 Efficiency: Local Lower A Posteriori Bound (n = 1) (Verfürth 89) Local dual norms: for v H 1 0 (ω) we have R, v = B(u U, v) α u U V v V R H 1 (ω) α u U V Interior residual: take ω = T T and note R, v = rv. Then T R H 1 (T ) = r H 1 (T ) Overestimation: Poincaré inequality yields r H 1 (T ) h T r L (T ) rv r L (T ) v L (T ) h T r L (T ) v L (T ) T Pw constant r: Let η H0 1 (T ), T T η, η L (T ) h 1 T. Then r L (T ) r(rη) r H 1 (T ) r L (T ) η L (T ) T h 1 T r H 1 (T ) r L (T ) h T r L (T ) r H 1 (T )

38 Efficiency: Local Lower A Posteriori Bound (n = 1) (Verfürth 89) Local dual norms: for v H 1 0 (ω) we have R, v = B(u U, v) α u U V v V R H 1 (ω) α u U V Interior residual: take ω = T T and note R, v = rv. Then T R H 1 (T ) = r H 1 (T ) Overestimation: Poincaré inequality yields r H 1 (T ) h T r L (T ) rv r L (T ) v L (T ) h T r L (T ) v L (T ) T Pw constant r: Let η H0 1 (T ), T T η, η L (T ) h 1 T. Then r L (T ) r(rη) r H 1 (T ) r L (T ) η L (T ) T h 1 T r H 1 (T ) r L (T ) h T r L (T ) r H 1 (T )

39 Lower A Posteriori Bound (Continued) Oscillation of r: h T r r T L (T ) with meanvalue r T. Then h T r L (T ) R H 1 (T ) + h T r r T L (T ) Data oscillation: if A is pw constant, then r = f and h T r r T L (T ) = h T f f T L (T ) = osc T (f, T ) Oscillation of j: likewise h S j j S L (S) with meanvalue j S and h 1/ S j L (S) R H 1 (ω S ) + h 1/ S j j S L (S) + h S r L (ω S ) where ω S = T 1 T with T 1 T = S and T 1, T T. Local lower bound: let ω T = S T ω S and the local oscillation be osc T (U, ω T ) := h(r r) L (ω T ) + h 1/ (j j) L ( T ). Then E T (U, T ) α (u U) L (ω T ) + osc T (U, ω T ).

40 Lower A Posteriori Bound (Continued) Oscillation of r: h T r r T L (T ) with meanvalue r T. Then h T r L (T ) R H 1 (T ) + h T r r T L (T ) Data oscillation: if A is pw constant, then r = f and h T r r T L (T ) = h T f f T L (T ) = osc T (f, T ) Oscillation of j: likewise h S j j S L (S) with meanvalue j S and h 1/ S j L (S) R H 1 (ω S ) + h 1/ S j j S L (S) + h S r L (ω S ) where ω S = T 1 T with T 1 T = S and T 1, T T. Local lower bound: let ω T = S T ω S and the local oscillation be osc T (U, ω T ) := h(r r) L (ω T ) + h 1/ (j j) L ( T ). Then E T (U, T ) α (u U) L (ω T ) + osc T (U, ω T ).

41 Lower A Posteriori Bound (Continued) Oscillation of r: h T r r T L (T ) with meanvalue r T. Then h T r L (T ) R H 1 (T ) + h T r r T L (T ) Data oscillation: if A is pw constant, then r = f and h T r r T L (T ) = h T f f T L (T ) = osc T (f, T ) Oscillation of j: likewise h S j j S L (S) with meanvalue j S and h 1/ S j L (S) R H 1 (ω S ) + h 1/ S j j S L (S) + h S r L (ω S ) where ω S = T 1 T with T 1 T = S and T 1, T T. Local lower bound: let ω T = S T ω S and the local oscillation be osc T (U, ω T ) := h(r r) L (ω T ) + h 1/ (j j) L ( T ). Then E T (U, T ) α (u U) L (ω T ) + osc T (U, ω T ).

42 Lower A Posteriori Bound (Continued) Higher order: we expect osc T (U, ω T ) (u U) L (ω T ) as h T 0. Marking: if E T (U, T ) (u U) L (ω T ) and E T (U, T ) is large relative to E T (U), then T contains a large portion of the error. To improve the solution U effectively, such T must be split giving rise to a procedure that tries to equidistribute errors. Global lower bound: we have E T (U) α u U V + osc T (U) where osc T (U) = h(r r) L (Ω) + h 1/ (j j) L (Γ). Discrete local lower bound (Dörfler 96, Morin, N, Siebert 00): E T (U, T ) α (U U) L (ω T ) + osc T (U, ω T ). provided the interior of T and each of its sides contain a node of T T (interior node property).

43 Lower A Posteriori Bound (Continued) Higher order: we expect osc T (U, ω T ) (u U) L (ω T ) as h T 0. Marking: if E T (U, T ) (u U) L (ω T ) and E T (U, T ) is large relative to E T (U), then T contains a large portion of the error. To improve the solution U effectively, such T must be split giving rise to a procedure that tries to equidistribute errors. Global lower bound: we have E T (U) α u U V + osc T (U) where osc T (U) = h(r r) L (Ω) + h 1/ (j j) L (Γ). Discrete local lower bound (Dörfler 96, Morin, N, Siebert 00): E T (U, T ) α (U U) L (ω T ) + osc T (U, ω T ). provided the interior of T and each of its sides contain a node of T T (interior node property).

44 Lower A Posteriori Bound (Continued) Higher order: we expect osc T (U, ω T ) (u U) L (ω T ) as h T 0. Marking: if E T (U, T ) (u U) L (ω T ) and E T (U, T ) is large relative to E T (U), then T contains a large portion of the error. To improve the solution U effectively, such T must be split giving rise to a procedure that tries to equidistribute errors. Global lower bound: we have E T (U) α u U V + osc T (U) where osc T (U) = h(r r) L (Ω) + h 1/ (j j) L (Γ). Discrete local lower bound (Dörfler 96, Morin, N, Siebert 00): E T (U, T ) α (U U) L (ω T ) + osc T (U, ω T ). provided the interior of T and each of its sides contain a node of T T (interior node property).

45 Outline Piecewise Polynomial Interpolation in Sobolev Spaces Model Problem and FEM FEM: A Posteriori Error Analysis Surveys

46 Surveys R.H. Nochetto Adaptive FEM: Theory and Applications to Geometric PDE, Lipschitz Lectures, Haussdorff Center for Mathematics, University of Bonn (Germany), February 009 (see R.H. Nochetto, K.G. Siebert and A. Veeser, Theory of adaptive finite element methods: an introduction, in Multiscale, Nonlinear and Adaptive Approximation, R. DeVore and A. Kunoth eds, Springer (009), R.H. Nochetto and A. Veeser, Primer of adaptive finite element methods, in Multiscale and Adaptivity: Modeling, Numerics and Applications, CIME Lectures, eds R. Naldi and G. Russo, Springer (to appear).