An optimal adaptive finite element method Rob Stevenson Korteweg-de Vries Institute for Mathematics Faculty of Science University of Amsterdam
Contents model problem + (A)FEM newest vertex bisection convergence of AFEM quasi-optimal computational complexity 1/16
Optimal adaptive finite element methods Model problem: Poisson, 2D, newest vertex bisection. [Generalizations: A with A symm. pos. def. piecewise constant, any space dimension n, red-refinements.] Given f H 1 (Ω), find u H 1 0(Ω), a(u, v) := := a(, ) 1 2. Ω u v = Ω fv =: f(v) (v H 1 0(Ω)). τ conforming part. of Ω into triangles, V τ = H 1 0(Ω) T τ P d 1(T ). Find u τ V τ, a(u τ, v τ ) = f(v τ ) (v τ V τ ). AFEM: GALSOLVE, compute a post. error est, MARK, REFINE. 2/16
Initial, conforming partition τ 0 Newest vertex bisection τ := REFINE[τ, M] : Bisect all T M τ a few times. Complete. Th 1 (Binev, Dahmen, DeVore 04). With suitable assignment of newest vertices in initial mesh, #τ K #τ 0 K 1 i=0 #M i (unif. in K). (can be generalized to any space dimension [St. 08]) 3/16
Perspective: Approximation classes A s := {u H 1 0(Ω) : u A s := sup N N s inf {τ P:#τ #τ 0 N} u u τ H 1 < }, [i.e. u u τ H 1 ε generally requires #τ #τ 0 ε 1/s u 1/s A s. ] When P unif. refs, then u A (d 1)/n = u H d (Ω), i.e., α u L 2 (Ω), α d With P all partitions created by newest vertex bisection: u A (d 1)/n = u B d q(l p (Ω)), any p > ( d 1 n + 1 2 ) 1 i.e., α u L p (Ω) α d ([Binev, Dahmen, DeVore, Petrushev 02]) Regul. th: For suff sm f, n = 2, u in such a Besov space for any d ([Dahlke, DeVore 97]). 4/16
A posteriori error estimator η T (f, u τ ) := diam(t ) 2 f + u τ 2 L 2 (T ) + diam(t ) u τ n T 2 L 2 ( T ), Th 2 (Babu ska, Rheinboldt 78; Verfürth 96). u u τ C 1 E(τ, f, u τ ) := C 1 [ η T (f, u τ )] 1 2. T τ Proof. u u τ = sup v H 1 0 (Ω) a(u u τ,v) v. a(u u τ, v) = a(u u τ, v v τ ) = = T τ T (f + u τ )(v v τ ) Ω T f(v v τ ) a(u τ, v v τ ) u τ n(v v τ ). Th 3 (St. 06, Cascon,Kreuzer,Nochetto,Siebert 07). σ τ, R τ σ := {T τ : T σ}, u σ u τ C 1 [ η T (f, u τ )] 1 T R τ σ Note #R τ σ #σ #τ. 5/16 2.
A posteriori error estimator We call σ τ a full refinement with respect to T τ, when T and all its neighbours in τ, as well as all faces of T contain a vertex of σ in their interiors. T τσ Th 4 (Morin, Nochetto, Siebert 00). Let f T τ P d 2(T ), σ τ full ref w.r.t. T τ. Then η T (f, u τ ) T {T } {neighbours} u σ u τ 2 H 1 ( T ) 6/16
Corol 5. σ τ full ref w.r.t. T M τ. Then c 2 [ η T (f, u τ )] 2 1 uσ u τ (not true for any f L 2 (Ω)) T M In part, c 2 E(τ, f, u τ ) u u τ. AFEM converges (Dörfler 96, Morin, Nochetto, Siebert 00) MARK[τ, f, u τ ] M: Let θ (0, 1]. Select smallest M τ s.t. [ η T (τ, f, u τ )] 1 2 θ E(τ, f, uτ ). T M REFINE[τ, M] σ: Construct smallest (conforming) σ τ that is a full refinement w.r.t. all T M. u u τ 2 = u u σ 2 + u σ u τ 2 u σ u τ c 2 θe(τ, f, u τ ) c 2θ C 1 u u τ u u σ (1 c2 2θ 2 ) 2 1 u uτ C 2 1 u τ u u σ 7/16 V σ
AFEM converges with optimal rate [St 06] ([Binev,Dahmen,DeVore 04] using coarsening) AFEM[f, ε] [τ m, u τm ] % let f T τ 0 P d 2 (T ) compute Gal sol u τ0 V τ0 ; k := 0 while C 1 E(τ k, f, u τk ) > ε do M k := MARK[τ k, f, u τk ] τ k+1 := REFINE[τ k, M k ] compute Gal sol u τk+1 V τk+1 k := k + 1 end do m := k Th 6. If θ < c 2 C 1, then u u τm ε, #τ m #τ 0 m 1 k=0 #M k #M k inf { #ρ #τ 0 : u u ρ [1 C2 1 θ2 ] 1 2 u u c 2 τk } 2 [ [1 C2 1 θ2 ] 1 2 u u c 2 τk ] 1/s 1/s u A s when u As 2. #τ m #τ 0 m 1 k=0 u u τk 1/s u 1/s A s u u τ m 1 1/s u 1/s A s ε 1/s u 1/s A s. 8/16
AFEM converges with opt rate Pr. Th. 6. Let σ τ k s.t. u u σ [1 C2 1 θ2 ] 1 2 u u c 2 τk. Th.3 shows 2 C 2 1 T R τk σ η T (f, u τk ) u σ u τk 2 = u u τk 2 u u σ 2 i.e., [ C2 1θ 2 c 2 u u τk 2 C1θ 2 2 E(τ k, f, u τk ) 2, 2 η T (f, u τk )] 1 2 θe(τk, f, u τk ). T R τk σ M := MARK[τ k, f, u τk ] is smallest set with this prop, so #M #R τk σ #σ #τ k. For arb. ρ with u u ρ [1 C2 1 θ2 ] 1 2 u u c 2 τk, take σ := τ k ρ. Then 2 #M #σ #τ k #ρ #τ 0. 9/16
General right hand sides and inexact solves RHS[τ, f, δ] [σ, f σ ] % In: τ a partition, f H 1 (Ω) and δ > 0. % Out: f σ T τ P d 2(T ), where σ = τ, or, if necessary, σ τ, % such that f f σ H 1 (Ω) δ. If u A s, cost of RHS will not dominate if c f s.t. #σ #τ c 1/s f δ 1/s, and cost #σ. Such a pair (f, RHS) is called s-optimal. For suff. smooth f, s-optimality with s = d n (> d 1 n ) can be realized. GALSOLVE[τ, f τ, u (0) τ, δ] w τ % In: τ, f τ (S τ ), and u (0) τ S τ. % Out: w τ S τ with u τ w τ H 1 (Ω) δ. % The call should require max{1, log(δ 1 u τ u (0) τ H 1 (Ω))}#τ ops. 10/16
AFEM has opt compl AFEM[f, ε] [τ m, w τm ] % ω, β > 0 suff small constants. w τ0 := 0; k := 0; δ 0 f H 1 (Ω) do do δ k := δ k /2 [τ k, f τk ] := RHS[τ k, f, δ k /2] w τk := GALSOLVE[τ k, f τk, w τk, δ k /2] if η k := (2 + C 1 c 1 2 )δ k/2 + C 1 E(τ k, f τk, w τk ) ε then stop endif until δ k ωe(τ k, f τk, w τk ). M k := MARK[τ k, f τk, w τk ] τ k+1 := REFINE[τ k, M k ] w τk+1 := w τk, δ k+1 := 2βη k, k := k + 1 enddo m := k Th 7. u w τm H 1 (Ω) ε. If u A s, and (f, RHS) is s-optimal, then both #τ m and work ε 1/s ( u 1/s A s + c1/s f ). 11/16
Realizations of RHS ex. f L 2 (Ω). Q σ being L 2 -orth.proj. onto T σ P d 2(T ). f Q σ f H 1 (Ω) T σ vol(t ) 2 n f Q T f 2 L 2 (T ) =: osc(f, σ) Given θ (0, 1), run MARK and REFINE algorithm on osc until δ. Is quasi-optimal in the sense that whenever f Ās := {f L 2 (Ω) : sup N N s inf osc(f, τ) < }, {τ:#τ #τ 0 N} then (f, RHS) is s-optimal (assuming T fp d 2 exact in O(1) operations) So greedy works (thanks to factor vol(t ) 2 n). 12/16
Realizations of RHS ex. General d, n > 1. ( 1 2 + 1 n ) 1 < q 2, f L q (Ω) ( H 1 (Ω)). Generalization of previous case with osc(f, σ) := T σ vol(t ) 1 2 q + 2 n inf p P d 2 (T ) f p 2 L q (T ). ex. n = 2, d = 2, f(v) = K v, K a smooth curve. RHS[τ, f, δ] [σ, f σ]: Refine those T that intersect K until their diameters δ 2. (f σ ) T := length(k T ) vol(t ). s-optimal for s = 1 2 (=d 1 n ). 13/16
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[6] R.S. Optimality of a standard adaptive finite element method. Found. Comput. Math., 7(2):245 269, 2007. [7] R.S. The completion of locally refined simplicial partitions created by bisection. Math. Comp., 77:227 241, 2008. [8] R. Verfürth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, 1996. 15/16
Extensions [Cascon,Kreuzer,Nochetto,Siebert 07]: One bisection of marked cells suffices (no interior node). Marking for reducing osc can be omitted, assuming f L 2 (Ω), exact integration, and with exact solving of Gal systems. L. Chen, M.J. Holst, and J. Xu. Convergence and optimality of adaptive mixed finite element methods. Accepted by Mathematics of Computation, 2007. Y. Kondratyuk, R. S. An optimal adaptive algorithm for the Stokes problem. To appear in SIAM J. Numer. Anal. 16/16