Convergence of the Adaptive Finite Element Method Carsten Carstensen Department of Mathematics, Humboldt-Universit¨ at zu Berlin http://www.math.hu-berlin.de/˜cc/ MATHEON, DFG Research Center Chemnitz FEM Symposium C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 1 / 32
Outline Introduction: UFEM&AFEM, References AFEM: Algorithm for Energy Minimization Short History of Arguments to Prove Error Reduction Convergence Theory for Convex Minimisation Applications and Examples Thanks to sponsors DFG, FWF, EPSRC and to collaborators S. Bartels, R. Hoppe, A. Orlando, J. Valdman C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 2 / 32
Introduction UFEM 1 Universal algorithm (uniform for all data: RHSs etc.) 2 T ℓ +1 := red ( T ℓ ) 3 Convergence from lim ℓ →∞ � h ℓ � L ∞ (Ω) = 0, but convergence can be arbitrarily bad (There is always some disaster RHS) AFEM 1 Specialised ”adapted” feedback algorithm (for ONE set of data) 2 T ℓ +1 generated from T ℓ , u ℓ , all data, plus extra computations 3 Convergence is open since � h ℓ � L ∞ (Ω) �→ 0 is not guaranteed a priori! Aims at error reduction or some other kind of convergence control C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 3 / 32
Books on a posteriori FE error control Eriksson-Estep-Hansbo-Johnson (1995) Verf¨ urth (1996) Ainsworth-Oden (2000) Babuska-Strouboulis (2001) Bangerth-Rannacher (2003) Neittaanm¨ aki-Repin (2004) ... and inside no theorem on AFEM convergence! (Except early results by Babuska et al. in 1D.) C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 4 / 32
Papers on Convergence of AFEM in 2D W. D¨ orfler : A convergent adaptive algorithm for Poisson’s equation. SIAM Journal on Numerical Analysis 33 (1996) 1106–1124. P. Morin , RH. Nochetto , and KG. Siebert : Local problems on stars: a posteriori error estimation, convergence, and performance. Mathematics of Computation 72 (2003) 1067–1097 AND Convergence of adaptive finite element methods. SIAM Review 44 (2003) 631–658. A. Veeser (2002) : Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math., 92 , 4, 743–770. P. Binev , W. Dahmen , and R. DeVore : Adaptive Finite Element methods with Convergence Rates. Num. Math., 97 (2) 219–268, (2004). R. Stevenson : Optimality of AFEM, preprint 2005. C , C-Hoppe , Braess-C-Hoppe C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 5 / 32
(AFEM) Input: coarse mesh T 0 SOLVE For ℓ = 0 , 1 , 2 , . . . ESTIMATE MARK REFINE Output: Sequence of nested discrete spaces ∞ � V ℓ ⊆ V = W 1 , p (Ω; R m ) V 0 ⊆ V 1 ⊆ V 2 ⊆ . . . ⊆ 0 ℓ =0 with associated stress approximations ( σ ℓ ) ℓ ∈ N 0 . Task: Design an (AFEM) with lim ℓ →∞ � σ − σ ℓ � L q = 0. C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 6 / 32
SOLVE Given shape regular triangulation T ℓ , define V ℓ := P 1 ( T ℓ ; R m ) ∩ V , compute some minimizer with Newton-Raphson scheme for � � W ( Dv ℓ ) dx − f · v ℓ dx for all v ℓ ∈ V ℓ . E ( v ℓ ) := Ω Ω Compute discrete stress σ ℓ := DW ( Du ℓ ) ∈ P 0 ( T ℓ ; R m × n ). W : R m × n → R with p th order growth causes V = W 1 , p (Ω; R m ), i.e., 0 gradients in L p and stresses in dual L p ′ , 1 / p + 1 / p ′ = 1. Remark: For class of degenerate convex minimization problems with p , p ′ , q , r , s , t , u and u ℓ are non-unique, σ and σ ℓ are unique s.t. � σ − σ ℓ � L r / t � min � u − v ℓ � W 1 , p . v ℓ ∈ V ℓ C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 7 / 32
ESTIMATE Given interior edge E = ∂ T + ∩ ∂ T − ∈ E ℓ , compute [ σ ℓ ] := σ ℓ | T + − σ ℓ | T − , η ( ℓ ) := diam( E ) 1 / p ′ � [ σ ℓ ] · ν E � L p ′ ( E ) E and set p ′ � 1 / p ′ � � η ( ℓ ) η ℓ := . E E ∈E ℓ ν E Theorem (C 2006). There holds T − � σ − σ ℓ � r L r / t (Ω) � η ℓ + osc ℓ T + E for oscillation osc p ′ � osc( f , ω z ) p ′ . ℓ := z ∈K ℓ For each inner node z with nodal basis function ϕ z ∈ V ℓ , the patch reads ω z := { x ∈ Ω : 0 < ϕ z ( x ) } and osc( f , ω z ) := diam( ω z ) � f − f ω z � L p ′ ( ω z ) . C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 8 / 32
MARK Bulk criterion with greedy algorithm. Sort E ℓ = { E 1 , . . . , E N } in list ( E 1 , . . . , E N ) s.t. η ( ℓ ) E 1 ≤ η ( ℓ ) E 2 ≤ . . . ≤ η ( ℓ ) E N and set k maximal with ℓ ≤ η ( ℓ ) p ′ + . . . + η ( ℓ ) p ′ Θ η p ′ E N . E k Then, for fixed 0 < Θ ≤ 1, M ℓ := { E k , E k +1 , . . . , E N } satisfies p ′ η p ′ η ( ℓ ) � . ℓ � E E ∈M ℓ Monitor osc ℓ +1 to achieve at least lim ℓ →∞ osc ℓ = 0. C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 9 / 32
REFINE Use concept of reference edge (i.e. opposite side of newest local vertex) E ( T ) for T = conv { E , F , G } and E , F , G ∈ E ℓ . Closure algorithm : Given shape-regular triangulation T := T ℓ and subset M := M ℓ of edges E ℓ in T ℓ , repeat (a)-(b) until T = ∅ : � � (a) Choose T ∈ T with E ( T ) ∩ M � = ∅ AND E ( T ) �∈ M and stop if there is no such T . � � T � = ∅ E ( T ) �∈ M (b) While AND do ( M := M ∪ { E ( T ) } , T := T \ { T } , T := N ( T )). Then M := closure( T ℓ , M ℓ ). Theorem (Bolte-C 2005 + ). M is minimal with M ℓ ⊆ M and ∀ E ∈ M ∀ T ∈ T ℓ ( E ) E ( T ) ∈ M . C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 10 / 32
Red-Green-Blue Refinement with Reference Edge Reference edge E ( T ) is bottom line and all M ( T ) are bisected via red(T) green(T) blue left blue right bisect5(T) bisec3(T) Inner Node Property achieved via bisect5(T) for at least on neighbouring triangle of each edge in M ℓ . C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 11 / 32
Mesh-Refinement Maintain shape regularity. On coarse K ∈ T 0 , T ℓ | K is affine picture of reference triangle with solely right isosceles triangles. T ℓ allows for ℓ -independent H 1 -stable L 2 -projections (C 2004). Binev-Dahmen-DeVore 2004, Bolte-C 2005 + show ℓ � card( T ℓ \ T 0 ) � card( M j ) . j =1 C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 12 / 32
Error Reduction Without Inner Node Property? A counterexample to error reduction for W = ψ ( | · | ) and nearest-vertex-bisection on regular polygon: T 0 T 1 Theorem (Bartels-C 2006 + ). Suppose Ω = � T 0 = � T 1 is regular polygon, decomposed in regular triangulations T 0 and T 1 . Then, σ 0 = σ 1 . C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 13 / 32
History of Arguments to Prove Error Reduction | and solution u ∈ W 1 , 2 (Ω; R m ) Linear elliptic PDE with energy norm | � · � 0 (and stress field σ ) respective discrete solution u ℓ (and σ ℓ ): (a) Reliability of error estimator [Rodriguez 94, C-Verf¨ urth SINUM 99] � | [ Du ℓ ] | 2 ds + osc( f ; T ℓ ) 2 | 2 � � | � u − u ℓ � h E E E ∈E ℓ (b) Bulk criterion [D¨ orfler SINUM 96] � | 2 � | [ Du ℓ ] | 2 ds + osc( f ; T ℓ ) 2 � | � u − u ℓ � h E E E ∈M ℓ (c1) Discrete local efficiency [D¨ orfler SINUM 96] for E ∈ M ℓ � | [ Du ℓ ] | 2 ds � | | ( ω E ) 2 + h 2 E � f � 2 h E � u ℓ +1 − u ℓ � L 2 ( ω E ) E C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 14 / 32
Cont. Arguments to Prove Error Reduction (c2) Refined discrete local efficiency [Nochetto et al., Veeser] � | [ Du ℓ ] | 2 ds � | | ( ω E ) 2 + h 2 E � f − f E � 2 h E � u ℓ +1 − u ℓ � L 2 ( ω E ) E (d) Finite overlap in (b)&(c2) yields | 2 ≤ C 1 | | 2 + C 2 osc( f ; T ℓ ) 2 | � u − u ℓ � � u ℓ +1 − u ℓ � (e) Galerkin orthogonality | 2 = | | 2 − | | 2 | � u ℓ +1 − u ℓ � � u − u ℓ � � u − u ℓ +1 � (f) Finish by rearranging | 2 ≤ ( C 1 − 1) | | 2 + C 2 osc( f ; T ℓ ) 2 C 1 | � u − u ℓ +1 � � u − u ℓ � and division by C 1 | 2 ≤ (1 − C − 1 | 2 + C − 1 1 C 2 osc( f ; T ℓ ) 2 | � u − u ℓ +1 � 1 ) | � u − u ℓ � C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 15 / 32
Substitute of Inner Node Property BAD OK Suppose free node z in T ℓ with patch ω z and edges E ( z ) in T ℓ | ω z . Assume all edges in E ( z ) are bisected but NOT all triangles in T ℓ ( ω z ) with bisec3 and reference edge on ∂ω z . Then � η 2 | 2 (Ω z ) + � osc( f , ω y ) 2 E � | � u ℓ +1 − u ℓ � y ∈ ω z ∩K E ∈E ( z ) C. Carstensen (Humboldt) Convergence of AFEM Chemnitz FEM 16 / 32
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