Axioms of Adaptivity (AoA) in Lecture 2 (sufficient for optimal convergence rates)

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1 Axioms of Adaptivity (AoA) in Lecture 2 (sufficient for optimal convergence rates) Carsten Carstensen Humboldt-Universität zu Berlin 2018 International Graduate Summer School on Frontiers of Applied and Computational Mathematics Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

2 Contents L2 1 Reduction Property 2 Plain Convergence 3 Linear Convergence 4 Quasimonotonicity 5 Comparison Lemma 6 Repetition on Optimal Rates 7 Separate Marking 8 Alternative Adaptive LS FEM 4 Laplace Open-Access Reference: C-Feischl-Page-Praetorius: AoA. Comp Math Appl 67 (2014) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

3 Reduction Property Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

4 With Dörfler Marking (A1)+(A2)=(A12) Abbreviate T := T l and T := T l+1 in (A1)-(A2) with η(k) := η l (K) for K T resp. η(t ) := η l+1 (T ) for T T and δ := δ(t, T ). Then (A1)-(A2) read η( T T ) η(t T ) Λ 1 δ η( T \ T ) ϱ 2 η(t \ T ) + Λ 2 δ Underlying sum convention η 2 (M) := M M η2 (M) Since marked T are refined, Dörfler marking leads to Θη 2 (T ) η 2 (T \ T ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

5 Weighted geometric-arithmetic mean inequality for a, b R reads (a + b) 2 (1 + λ) a 2 + (1 + 1/λ) b 2 for all λ > 0 (with equality if a, b > 0 for the minimizing choice of λ) Application to (A1) shows for any λ > 0 η 2 ( T T ) (1 + λ)η 2 (T T ) + (1 + 1/λ)Λ 2 1δ 2 Application to (A2) shows for any µ > 0 η 2 ( T \ T ) ϱ 2 2(1 + µ)η 2 (T \ T ) + (1 + 1/µ)Λ 2 2 δ 2 The sum gives η 2 ( T ) = η 2 ( T T ) + η 2 ( T \ T ) on LHS Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

6 Hence η 2 ( T ) (1 + λ)η 2 (T T ) + ϱ 2 2(1 + µ)η 2 (T \ T ) + ( (1 + 1/λ)Λ (1 + 1/µ)Λ 2 ) 2 2 δ }{{} Λ 12 Recall 0 < Θ 1 and 0 < ϱ 2 < 1 so there exist 0 < µ < ϱ and 0 < λ < Θ 1 (1 + µ)ϱ2 2 1 Θ Theorem (A12) (A1)-(A2) and Dörfler marking imply η 2 ( T ) ϱ 12 η 2 (T ) + Λ 12 δ 2 with ϱ 12 < 1 and Λ 12 < for any such choice of λ, µ Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

7 Proof of (A12) Recall η 2 ( T ) (1 + λ)η 2 (T T ) + ϱ 2 2(1 + µ)η 2 (T \ T ) + Λ 12 δ 2 Recall η 2 (T T ) = η 2 (T ) η 2 (T \ T ) and rewrite η 2 ( T ) Λ 12 δ 2 (1 + λ)η 2 (T T ) + ϱ 2 2(1 + µ)η 2 (T \ T ) ( ) = (1 + λ)η 2 (T ) + ϱ 2 2(1 + µ) (1 + λ) η 2 (T \ T ) Since ϱ 2 2 (1 + µ) < 1 < 1 + λ, the factor in brackets is 0 and Dörfler marking with Θη 2 (T ) η 2 (T \ T ) leads to Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

8 η 2 ( T ) Λ 12 δ 2 ( ) (1 + λ)η 2 (T ) + ϱ 2 2(1 + µ) (1 + λ) η 2 (T \ T ) ( ) (1 + λ)(1 Θ) + Θϱ 2 2(1 + µ) η 2 (T ) }{{} ϱ 12 The proof concludes with ϱ 12 < 1 iff λ < Θ 1 (1+µ)ϱ2 2 1 Θ calculation) (by a minor NB. λ, µ small reduce ϱ 12 but increase Λ 12 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

9 Repeat (A12) Abbreviate T := T l and T := T l+1 in (A1)-(A2) with η(k) := η l (K) for K T resp. η(t ) := η l+1 (T ) for T T and δ := δ(t, T ) =: δ l,l+1. Recall there exist 0 < µ < ϱ and 0 < λ < Θ 1 (1 + µ)ϱ2 2 1 Θ Theorem (A12) (A1)-(A2) and Dörfler marking imply ηl+1 2 ϱ 12 ηl 2 + Λ 12δl,l+1 2 with ϱ 12 < 1 and Λ 12 < for any such choice of λ, µ Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

10 Plain Convergence Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

11 Convergence η k 0 as k Theorem (plain convergence) (A12), (A4), and Λ := (1 + Λ 12 Λ 4 )/(1 ϱ 12 ) < imply ηk 2 Λη2 l for all l N 0. k=l Proof. Recall (A12) as η 2 k+1 ϱ 12η 2 k + Λ 12δ 2 k,k+1. Then l+m k=l l+m+1 ηk 2 k=l η 2 k η2 l + ϱ 12 l+m k=l η 2 k + Λ 12 l+m k=l Recall (A4) k=l δ2 k,k+1 Λ 4ηl 2 for the last term and utilize ϱ 12 < 1 to conclude the proof δ 2 k,k+1 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

12 Linear Convergence Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

13 R-Linear Convergence Uniformly on Each Level Theorem (A12),(A4),Λ, and q := 1 1/Λ < 1 imply ηl+m 2 qm Λ ηl 2 for all l, m N 0. Proof. Plain convergence gives ξl 2 := ηk 2 Λ η2 l < k=l and so Λ 1 ξl 2 η2 l = ξ2 l ξ2 l+1 This proves ξl+1 2 qξ2 l (for each l N 0 ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

14 Recall ξ 2 l+1 qξ2 l for all l N 0 Mathematical induction leads to ξ 2 l+m qm ξ 2 l for all m N 0 This and ξ 2 l := k=l η2 k Λ η2 l show η 2 l+m ξ2 l+m qm ξ 2 l qm Λη 2 l R-linear convergence uniformly on each level implies via a geometric series l 1 k=0 η 1/s k η 1/s l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

15 Quasimonotonicity Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

16 (QM) Estimator Quasimonotonicity Theorem (QM) (A1) (A3) and Λ mon := 1 + Λ Λ2 2 Λ 3 imply η( T ) Λ mon η(t ) for any refinement T of any T in T Proof. For any 0 < λ <, utilize (A1)-(A2) in the decomposition η 2 ( T ) = η 2 ( T T ) + η 2 ( T \T ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

17 η 2 ( T ) = η 2 ( T T ) + η 2 ( T \T ) ( (1 + λ) η 2 (T T ) + η 2 (T \ T ) ) }{{} η 2 (T ) + (1 + 1/λ)(Λ Λ 2 2)δ 2 (A3) reads δ 2 Λ 2 3 η2 (T ) and leads to η 2 ( T ) (1 + λ + (1 + 1/λ)(Λ Λ 2 2)Λ 2 3)η 2 (T ) }{{} Λ 2 mon Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

18 Comparison Lemma Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

19 Comparison Lemma Given 0 < κ < 1 and s > 0 with M := sup (N + 1) s min η(t ) < N N 0 T T(N) 0 < θ 0 < 1 l N 0 ˆT l T(T l ) s.t. (a) η( ˆT l ) κη(t l ) (b) κη l T l \ ˆT l s Λ mon M (c) θ 0 η 2 l η2 l (T l \ ˆT l ) Proof. (1) W.l.o.g. η l η(t l ) > 0. Then (QM) implies 0 < η 0 M Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

20 (2) Choose minimal N l N 0 s.t. (N l + 1) s κη l Λ mon M < N s l 1 (N l 1 because η l Λ 1 mon/m η 0 /M 1) (3) Set T l := T l T for T with T T(N l ) s.t. (N l + 1) s η(t ) M Quasimonotonicity and overlay control lead to (a), η( T l ) Λ mon M(N l + 1) s κη l and T l T l + N l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

21 Proof of (b)-(c) in Comparison Lemma (4) Proof of (b). Count triangles to verify T l \ T l T l T l N l < κ 1/s η 1/s l Λ 1/s monm 1/s from from (2) (5) Any T l T(T l ) with (a) allows for (c). Given 0 < µ < κ 2 1, (A1) plus (a) and (A3) imply η 2 l (T l T l ) (1+µ)η 2 ( T l, T l T l ) +(1+1/µ)Λ 2 1δ 2 (T l, T l ) (1 + µ)κ 2 η 2 l + (1 + 1/µ)Λ2 1Λ 2 3η 2 l (T l\ T l ) This and η 2 l = η2 l (T l T l ) + η 2 l (T l\ T l ) lead to (1 (1 + µ)κ 2 ) η 2 l (1 + (1 + 1/µ)Λ2 1Λ 2 3) η 2 l (T l\ T l ) NB. Θ 0 is a quotient that depends onκ and µ. Fix those paramaters. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

22 Repetition on Optimal Rates Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

23 Optimality Analysis at a Glance II R satisfies bulk criterion if θ A θ 0 thus M l R for optimal set M l of marked cells. AFEM utilizes almost minimal M l, whence M l M l R Set M := sup N N0 (1 + N) s min T T(N) η(t ) with (writing κ 1) R M 1/s η 1/s l Recall closure overhead control and combine with aforementioned estimates for l 1 l 1 T l T 0 M k M 1/s k=0 k=0 η 1/s k M 1/s η 1/s l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

24 Separate Marking Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

25 Adaptive Algorithm: SAFEM Input: T 0, 0 < κ, θ A 1, ρ B < 1 l = 0, 1, 2, 3,... Compute η 2 l (K) and µ2 (K) for all K T l if µ 2 l := µ2 (T l ) κη 2 l T l+1 := Dörfler marking(θ A, T l, η 2 l ) else T l+1 := T l data approximation(ρ B µ 2 l, T 0, µ 2 l ) Output: Sequence (T l ), (η l ), (µ l ) abreviate σ 2 l := η2 l + µ2 l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

26 Axioms (A1) (A4) Supposeρ 2 <1,Λ k < s.t. T T ˆT T (T ) R T s.t. T \ ˆT R R T \ ˆT η( ˆT, T ˆT ) η(t, T ˆT ) Λ 1 δ(t, ˆT ) (A1) η( ˆT, ˆT \ T ) ρ 2 η(t, T \ ˆT ) + Λ 2 δ(t, ˆT ) (A2) δ 2 (T, ˆT ) Λ 3 ( η 2 (T, R)+µ 2 (T ) ) + Λ 3 η 2 ( ˆT ) (A3) δ 2 (T k, T k+1 ) Λ 4 σl 2 for all l N 0 (A4) k=l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

27 Axioms (B1) (B2) Tol > 0 T Tol = data approx(tol, T 0, µ 2 ) T satisfies µ 2 (T Tol ) Tol and T Tol T 0 Λ 5 Tol 1/(2s) (B1) T T, ˆT T(T ) µ 2 ( ˆT ) Λ 6 µ 2 (T ) (B2) (B1) (B2) follow for Approx from subadditivity µ 2 ( ˆT (M)) := µ 2 (T ) Λ 6 µ 2 (M) (SA) K M T ˆT (K) Typical example TSA by Binev and DeVore but also Dörfler marking over sufficient levels could do [C-Rabus 2016] Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

28 Theorem (C-Rabus 2016) SAFEM leads to optimal convergence rates in total estimators provided (A1)-(A4), (B1)-(B2), 0 < θ A < θ 0 := 1 κλ2 1 Λ Λ 2 1 Λ 3 and κ < 1 ρ A Λ 6 1 plus Quasimonotonicity (e.g. for (Λ Λ2 2 ) Λ 3 < 1) T T ˆT T (T ) σ( ˆT ) Λ 7 σ(t ). Applications to div-lsfem and to MFEM with convergence rates in H(div, Ω) L 2 (Ω) in [C-Rabus 2016] Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

29 Alternative Adaptive LS FEM 4 Laplace Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

30 Alternative A Posteriori Error Control Estimator η and data approximation error µ in 2D η 2 (T, K) := (1 Π 0 )p LS 2 L 2 (K) + K 1/2 ( ) [p LS ] E τ E 2 L 2 (E) + [ u LS/ ν E ] E 2 L 2 (E\ Ω) E E(K) µ 2 (K) := f Π 0 f 2 L 2 (K) for K T [C-Park 2015] satisfy discrete reliability (A3) (for k = 0) (Proof with explicit Crouzeix-Raviart and Raviart-Thomas functions, discrete Helmholtz decomposition, mixed intermediate solutions still leaves extra term) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

31 References and Further Reading C, M. Feischl, M. Page, D. Praetorius, Axioms of adaptivity, Comput. Math. Appl., 67 (2014), pp C, H. Rabus. Axioms of adaptivity with separate marking for data resolution. SIAM J. Numer. Anal. 55(6) (2018) P. Bringmann, C, G. Starke: An adaptive least-squares FEM for linear elasticity with optimal conv rates, SIAM J. Numer. Anal. 56 (2018) P. Bringmann, C: h-adaptive least-squares FEMs for 2D Stokes eqns of any order with optimal conv rates, Comput. Math. Appl. 74 (2017) P. Bringmann, C: An adaptive least-squares FEM for Stokes eqns with optimal conv rates, Numer. Math. 135 (2017) C, E. J. Park, P. Bringmann: Convergence of natural adaptive least squares FEMs, Numer. Math. 136 (2017) C, E.-J. Park: Convergence and optimality of adaptive least squares FEMs, SIAM J. Numer. Anal. 53 (2015) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32

32 Prepare yourself for tomorrow Conforming P 1 FEM for Poisson Model Problem (weak form, H0 1(Ω), energy scalar product a(, ) := Ω dx, energy norm) Inverse estimates (for polynomials) Trace inequality (for Sobolev functions) Discrete trace inequality (for polynomials) Shape regularity (for triangles, simplices) Poincaré and Friedrichs inequality (for Sobolev functions) Equivalence of norms in finite dimensional vector spaces Scaling argument (for derivatives of Sobolev functions) Triangle inequality (in normed linear spaces) Cauchy inequality (in Hilbert spaces like L 2 or w.r.t. a(, )) C, F. Hellwig: Constants in Discrete Poincaré and Friedrichs Inequalities and Discrete Quasi-interpolation, CMAM (arxiv: ), Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai / 32