Axioms of Adaptivity (AoA) in Lecture 1 (sufficient for optimal convergence rates)

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1 Axioms of Adaptivity (AoA) in Lecture 1 (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 1 Shanghai / 29

2 EUROPEAN COMMUNITY ON COMPUTATIONAL METHODS IN APPLIED SCIENCES COMMUNICATED BY CARSTEN CARSTENSEN, CHAIRMAN OF THE ECCOMAS TECHNICAL COMMITTEE "SCIENTIFIC COMPUTING" RATE OPTIMALITY OF ADAPTIVE ALGORITHMS The overwhelming practical success of adaptive mesh-refinement in computational sciences and engineering has recently obtained a mathematical foundation with a theory on optimal convergence rates. This article first explains an abstract adaptive algorithm and its marking strategy. Secondly, it elucidates the concept of optimality in nonlinear approximation theory for a general audience. It thirdly outlines an abstract framework with fairly general hypotheses (A1) (A4), which imply such an optimality result. Various comments conclude this state of the art overview. All details and precise references are found in the open access article [C. Carstensen, M. Feischl, P. Page, D. Praetorius, Comput. Math. Appl. 67 (2014)] at j.camwa adaptive mesh-refining algorithm, where the marking is the essential decision for refinement and written as a list of ℳ cells (i.e. ℳ ) with some larger refinement-indicator. The refinement procedure then computes the smallest admissible refinement of the mesh (see Section 3) such that at least the marked cells are refined. The successive loops of those steps lead to the following adaptive algorithm, where the coarsest mesh 0 is an input data. Adaptive Algorithm Input: initial mesh 0 Loop: for 0, 1,2, do steps 1-4: 1. Solve: Compute discrete approximation U ( ). 2. Estimate: Compute refinement indicators ητ( ) for all T. 3. Mark: Choose set of cells to refine ℳ (see Section 4 for details). THE ALGORITHM 4. Refine: Generate new mesh +1 by refinement of at least all cells in ℳ (see Section 3 for details). Output: Meshes, approximations U ( ), and estimators η( ). THE OPTIMALITY Figure 1 displays a typical mesh for some adaptive 3D mesh-refinement of some L-shaped cylinder into tetrahedra with some global refinement as well as some local mesh-refinement towards the vertical edge along the re-entrant corner. The question whether this is a good mesh or not is an important issue in the mesh-design with many partially heuristic answers and approaches. We merely mention the coarsening techniques as in [Binev et al., 2004] when applied to the adaptive hp-fem with the crucial decision about h- or p-refinement. For the optimality analysis of the adaptive algorithm of Section 1, the The geometry of the domain Ω in some boundary value problem (BVP) is often specified in numerical simulations in terms of a triangulation (also called mesh or partition) which is a set of a large but finite number of cells (also called element-domains) Τ1,, ΤΝ. Based on this mesh, some discrete model (e.g., finite element method (FEM)) leads to some discrete solution U( ) which approximates an unknown exact solution u to the BVP. Usually, a posteriori error estimates motivate some computable error estimator The local contributions ητj ( ) serve as refinement-indicators in the Rate Optimality of Adaptive Algorithms C-Feischl-Praetorius ECCOMAS newsletter (electr.) 2014 pp Figure 1: Strongly adaptively refined surface triangulation 20 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

3 Contents 1 Introduction 2 Axioms at a Glance 3 Optimality Analysis Outline 4 Adaptive Mesh-Refining Open-Access Reference: C-Feischl-Page-Praetorius: AoA. Comp Math Appl 67 (2014) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

4 Introduction Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

5 Goals Identify minimal assumptions on problem and estimator η 2 l := η2 (T l ) := K T l η 2 l (K) with (ηl 2(K)) (η l(k)) 2 etc. to guarantee plain convergence η l η(t l ) 0 as l R-linear convergence η l η(t l ) 0 as l optimal convergence rates for estimators η l ( T l T 0 ) s optimal convergence rate for errors u u l ( T l T 0 ) s Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

6 Short History 1D results with Babuska et al in the Eighties Dörfler marking [Dörfler 1996] Convergence [Morin-Nochetto-Siebert 2000] Optimal rates Poisson problem [Binev-Dahmen-DeVore 2004] Optimal rates without coarsening [Stevenson 2007] NVB included [Cascon-Kreuzer-Nochetto-Siebert 2008] Integral equations and BEMs [Feischl et al. 2013], [Gantumur 2013] General boundary conditions [Aurada et al. 2013] General 2.-order lin. ellip. PDEs [Feischl et al. 2014] Axioms of Adaptivity (AoA) [C-Feischl-Page-Praetorius 2014] Instance optimality [Diening-Kreuzer-Stevenson 2015] AoA separate marking [C-Rabus 2017] Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

7 Adaptive Algorithm I - CAFEM Input: Initial triangulation T 0 and parameter 0 < θ A 1 l = 0, 1, 2, 3,... (until termination for some other reasons) Compute η l (K) for all K T l T l+1 :=Dörfler marking(θ A, T l, η 2 l ) Output: Sequence (T l ) and (η l ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

8 Adaptive Algorithm I - CAFEM Input: Initial triangulation T 0 and parameter 0 < θ A 1 l = 0, 1, 2, 3,... (until termination for some other reasons) Compute η l (K) for all K T l T l+1 :=Dörfler marking(θ A, T l, η 2 l ) Determine (almost) minimal set of marked cells M l T l s.t. θ A η 2 l η2 l (M l) := M M l η 2 l (M) Design minimal refinement T l+1 of T l with M l T l \ T l+1 Output: Sequence (T l ) and (η l ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

9 Admissible Triangulations Newest-vertex bisection (NVB) refinement strategy and initial triangulation T 0 specifies set T of all admissible triangulations T green(t ) blue R(T ) blue L(T ) bisec3(t ) NVB in any dimension Closure Overhead Control [Stevenson 2008 Math.Comp.] l 1 T l T 0 C BDV M j j=0 Overlay Control T l T ref + T 0 T l + T ref Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

10 Optimal Convergence Rates Axioms (A1) (A4) involve estimators 0 η(t ; K) < for all K T T distances 0 δ(t, ˆT ) < for all ˆT T(T ) The axioms imply rate-optimality for θ A 1 in that sup (1 + N) s N N 0 min T T(N) η(t ) sup l N 0 (1 + T l T 0 ) s η l η 2 l := η2 (T l ) and η 2 (T ) := η 2 (T, T ) := K T η 2 (T, K) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

11 sup (1 + N) s N N 0 min T T(N) η(t ) sup(1 + T l T 0 ) s η l N 0 }{{} l N l σ(t 1 ) σ(t 2 ) σ(t 3 ) 1 s Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

12 Poisson Model Problem (PMP) CFEM seeks u C P k (T ) C 0 (Ω) s.t. for all v C P k (T ) C 0 (Ω) u C v C dx = fv C dx Ω Ω CR-NCFEM seeks u CR CR0(T 1 ) s.t. for all v CR CR0(T 1 ) NC u CR NC v CR dx = fv CR dx Ω Ω RT-MFEM seeks p RT RT k (T ) and u RT P k (T ) s.t. p RT q RT dx + u RT div q RT dx = 0 Ω Ω Π 0 f + div p RT = 0 for all q RT RT k (T ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

13 Error estimator For all K T l ηl 2 (K):= K 2/n f + div p l 2 L 2 (K) + K 1/n [p l ] E 2 L 2 (E) for flux approximation p l from conforming FEM p l := u C non-conforming FEM p l := NC u CR mixed FEM p l := p RT E E(K) and jumps [p l ] E := (p l T+ ) (p l T ) on E = T + + T with appropriate modifications for E Ω Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

14 Axioms at a Glance Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

15 Axioms (A1) (A4) Suppose ρ 2 < 1, Λ 1,... Λ 4 s.t. T T ˆT T (T ) R with T \ ˆT R T R T \ ˆT s.t. 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(t, ˆT ) Λ 3 η(t, R) (A3) δ 2 (T k, T k+1 ) Λ 4 ηl 2 for all l N 0 (A4) k=l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

16 Optimality Analysis Outline Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

17 Optimality Analysis at a Glance I (A12) Estimator reduction η 2 l+1 ρ lη 2 l + Λ 12δ 2 l,l+1 (A12) and (A4) imply convergence from ηk 2 η2 l k=l and then l 1 k=0 η 1/s k η 1/s l (A12) and (A3) imply quasimonotonicity (QM) η( ˆT ) Λ 7 η(t ) Comparison lemma: l 0 < ϱ < 1 ˆT l T(T l ) θ 0 < 1 s.t. η( ˆT l ) ϱη(t l ) ϱη l R s sup (1 + N) s N N 0 min T T(N) η(t ) =: M θ 0 η 2 l η2 (T l, R) for R from (A3) for ˆT l T(T l ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

18 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 ϱ 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 j M 1/s j=0 j=0 η 1/s j M 1/s η 1/s l Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

19 Adaptive Mesh-Refining Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

20 Initial Triangulation in 2D Suppose T 0 is regular triangulation of the bounded polygonal Lipschitz domain Ω R 2 into triangles; given any K T 0, chose one of its three edges ref(k) as the refinement edge of K. Definition (IC) The regular triangulation with refinement edges (ref(k) : K T 0 ) satisfies (IC) if for any interior edge E = K + K E 0 (Ω) in T 0 with the two neighbouring triangles K + and K in T 0 it holds either ref(k + ) = E = ref(k ) or ref(k + ) E ref(k ). NB. (IC) allows for uniform bisections: If regular triangulation with refinement edges (ref(k) : K T 0 ) satisfies (IC) then, any uniform bisection (i.e. bisect every refinement edge and define the new refinement edges opposite of the new vertex) leads to a regular triangulation and it satisfies (IC). Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

21 Refinements of a Triangle and a Triangulation None green blue (left) blue (right) bisec3 Conventions: ref(k) drawn as base edge and the refinement edge is the hypotenuse in any right isosceles sub-triangle Definition (one-level refinement) Let T (resp. T ) be a regular triangulation of Ω into triangles with refinement edges (ref(t ) : T T ) (resp. (ref(t ) : T T )). Then T is called a one-level refinement of T if any coarse triangle K T is refined as depicted above into J(K) {0, 1, 2, 3, 4} sub-triangles T (K) = {T 1,..., T J(K) } s.t. T = K T T (K) is the collection of all of them Definition (refinement) T is called a refinement of T if there is a finite sequence of successive one-level refinements that begins with T and ends with T Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

22 Admissible Triangle and Admissible Triangulation Convention: Let T 0 be an initial triangulation (i.e. a regular triangulation of Ω into triangles with refinement edges (ref(t ) : T T 0 )) with (IC). Definition (admissible triangulation) The set T of admissible triangulations is the set of all refinements of T 0 Notation: T T(T 0 ) and, more general, T T(T ) abbreviates that T is a refinement of T T (NB. T 0 satisfies (IC) but i.g. T T does not) Definition (admissible triangle) T = {T T : T T} is the set of admissible triangles Some first observations on admissible triangles and their refinement edges are summarized in the following lemma: (a) level of a triangle, (b) successive bisections, (c) refinement edges are independent of the way the admissible triangle is created, (d) admissible triangles, (e) finite overlap of two admissible triangles implies inclusion, and (f) shape regularity Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

23 Elementary Properties 1 (a) For any admissible triangle T T there exists a unique K T 0 with T K and a level l(t ) N 0 of T with 2 l(t ) = K / T. (b) For any admissible triangle T T there exists a unique finite sequence of l(t ) admissible triangles T 0, T 1,..., T l(t ) T with T 0 = K T 0 and T l(t ) = T s.t. T k+1 bisec(t k ), i.e., T k+1 is generated in the bisection of T k, for all k = 0,..., l(t ) 1. (Consequently, k = l(t k ) for k = 0, 1,..., l(t ).) (c) Any admissible triangle T T of level l(t ) N 0 belongs to the partition bisec (l(t )) (T 0 ) obtained by l(t ) bisections of triangles in T 0, written T bisec (l(t )) (T 0 ). In particular, T = k N 0 bisec (k) (T 0 ). Remark: The refinement edges are highlighted in the initial triangulation T 0 and then inherited during the successive bisections. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

24 Elementary Properties 2 (d) The refinement edge of any admissible triangle is uniquely determined by the initial refinement edges (ref(k) : K T 0 ). In other words, suppose T 0, T 1,..., T L and T 0, T 1,..., T L are two sequences of successive one-level refinements of T 0 and they inherit the refinement edges during those successive refinements. Then the two definitions of the refinement edge of T T L T L coincide. (e) If the intersection int(t 1 T 2 ) of two admissible triangles T 1, T 2 T contains interior points, int(t 1 ) int(t 2 ), and if l(t 1 ) l(t 2 ), then T 2 T 1. (f) Given K T and T T, the set T (K) := {T T : T K} may be empty. But otherwise T (K) is a regular triangulation of K and is equal to the affine image of an admissible triangulation of T ref = conv({(0, 0), (1, 0), (0, 1)}) into right isosceles triangles (with hypothenuses as refinement edges) (g) There are at most 8 T 0 different interior angles in T. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

25 Lemma (levels of two triangles with a common edge) If T 0 satisfies (IC) and T 1, T 2 T T share a common edge T 1 T 2 = E, then exactly one case out of (A)-(D) occurs ref(t 1 ) E = ref(t 2 ) l(t 1 ) = l(t 2 ) 1 (A) ref(t 1 ) = E ref(t 2 ) l(t 1 ) = l(t 2 ) + 1 (B) ref(t 1 ) = E = ref(t 2 ) l(t 1 ) = l(t 2 ) (C) ref(t 1 ) E ref(t 2 ) l(t 1 ) = l(t 2 ) (D) (A) (B) (C) (D) T 2 T 2 T 1 T 1 T 1 T 1 T 2 T 2 Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

26 Overlay T 1 T 2 Definition (overlay) Given two regular triangulations T 1, T 2 T of Ω into admissible triangles, their overlay is T 1 T 2 := {T 1 T 2 : T 1 T 1, T 2 T 2 with int(t 1 ) int(t 2 ) } Lemma (overlays are regular) Given two regular triangulations T 1, T 2 T of Ω into admissible triangles, their overlay T 1 T 2 T 1 T 2 is a regular triangulation and T 1 T 2 + T 0 T 1 + T 2. Remark: An admissible triangulation is described as a forrest of to binary trees and then the overlay is the union of the two forrests. The extra point is that it is a regular triangulation. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

27 Procedure Refine Neighbour map N : T T is defined for T T by { K if ref(t ) E(K) for K T \ {T }, N(T ) := T if ref(t ) Ω Theorem For all M T T, T := refine(t, M) below computes a one-level refinement of T with M T \ T of minimal nodes. (Minimal nodes ˆN in the following sense: If T T is a regular triangulation in admissible triangles with M T \ T and the set Ñ of nodes, then N Ñ implies ˆN Ñ.) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

28 T := refine(t, M) Given input M T T then procedure refine(t, M) if M T set T 0 := M j = 0, 1, 2,... (until termination with output J := j) compute neighbour T j+1 := N(T j ) {T j1, T j2 } := bisec(t j ) T := (T \ {T j }) {T j1, T j2 } if j 1 and (ref(t j 1 ) = ref(t jk ) for one k {1, 2}) then T := (T \ {T jk }) bisec(t jk ) if T j+1 {T 0,..., T j } then terminate with J := j output T := refine(t, M) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

29 Properties of Refine The refine lemma. Any M T T and any T refine(t, M) \ T satisfy (3 l(t ))/2 l(t ) l(m) + 1 and dist(m, T ) D 2 with the universal constant D := max T T max T T diam(t ) 2 l(t )/2 1. The commuting lemma. Given any M 1, M 2 T T set T j := refine(t, M j ) and T jk := refine(t j, M k ) for {j, k} = {1, 2} Then T 12 = T 21 = T 1 T 2. Refinement in AFEM. On the level l N 0, the adaptive algorithm marks by a nonvoid set M l T l of cardinality m(l) := M l N and computes the refinement T l+1 first by an enumeration {M l,1,... M l,m(l) } = M l and second the steps (leave the second step out for m(l) = 1) T l+1,0 := T l T l+1,m+1 := refine(t l+1,m, M l,m+1 ) (m = 0,..., m(l) 2) T l+1 := refine(t l+1,m(l) 1, M l,m(l) ) Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29

30 Theorem of Binev-Dahmen-DeVore (2004) The previous mesh-refining algorithm allows a re-interpretation of the AFEM algorithm to compute a subsequence of triangulations (namely j = 0, j = N 1, j = N 1 + N 2,... and so j = N 1 + N N l 1 on the level l with N l := M l ) of the sequence T (0) := T 0 and for all j N 0 do Select M(j) T (j) and T (j + 1) := refine(t (j), M(j)). Theorem (Binev-Dahmen-DeVore) There exists a constant c BDV that depends only on T 0 such that T (j) T 0 c BDV j for all j N 0. It is remarkable that the choice of M(j) T (j) does not play any role. Carsten Carstensen (HU Berlin) Axioms of Adaptivity Lecture 1 Shanghai / 29