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 2018 1 / 32
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) 1195 1253 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 2 / 32
Reduction Property Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 3 / 32
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 2018 4 / 32
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 2018 5 / 32
Hence η 2 ( T ) (1 + λ)η 2 (T T ) + ϱ 2 2(1 + µ)η 2 (T \ T ) + ( (1 + 1/λ)Λ 2 1 + (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 2 2 1 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 2018 6 / 32
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 2018 7 / 32
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 2018 8 / 32
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 2 2 1 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 2018 9 / 32
Plain Convergence Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 10 / 32
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 2018 11 / 32
Linear Convergence Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 12 / 32
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 2018 13 / 32
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 2018 14 / 32
Quasimonotonicity Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 15 / 32
(QM) Estimator Quasimonotonicity Theorem (QM) (A1) (A3) and Λ mon := 1 + Λ 2 1 + 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 2018 16 / 32
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 2 1 + 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 2 1 + Λ 2 2)Λ 2 3)η 2 (T ) }{{} Λ 2 mon Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 17 / 32
Comparison Lemma Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 18 / 32
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 2018 19 / 32
(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 2018 20 / 32
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 2018 21 / 32
Repetition on Optimal Rates Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 22 / 32
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 2018 23 / 32
Separate Marking Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 24 / 32
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 2018 25 / 32
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 2018 26 / 32
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 2018 27 / 32
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 3 1 + Λ 2 1 Λ 3 and κ < 1 ρ A Λ 6 1 plus Quasimonotonicity (e.g. for (Λ 2 1 + 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 2018 28 / 32
Alternative Adaptive LS FEM 4 Laplace Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 29 / 32
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 2018 30 / 32
References and Further Reading C, M. Feischl, M. Page, D. Praetorius, Axioms of adaptivity, Comput. Math. Appl., 67 (2014), pp. 1195-1253. C, H. Rabus. Axioms of adaptivity with separate marking for data resolution. SIAM J. Numer. Anal. 55(6) (2018) 2644-2665 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 2018 31 / 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:1709.00577), 2017. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 2 Shanghai 2018 32 / 32