A MULTILEVEL PRECONDITIONER FOR THE INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD

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1 A MULTILEVEL PRECONDITIONER FOR THE INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD KOLJA BRIX, MARTIN CAMPOS PINTO, AND WOLFGANG DAHMEN Abstract. In tis article we present a multilevel preconditioner for interior penalty discontinuous Galerkin discretizations of second order elliptic boundary value problems tat gives rise to uniformly bounded condition numbers witout any additional regularity assumptions on te solution. Te underlying triangulations are only assumed to be sape regular but may ave anging nodes subject to certain mild grading conditions. A key role is played by certain decompositions of te discontinuous trial space into a conforming subspace and a non-conforming subspace tat is controlled by te jumps across te edges. AMS Subject Classification: 65F10, 65N55, 65N30 Key Words: Interior penalty metod, energy-stable splittings, admissible averaging operators, frames, multilevel Scwarz preconditioners, discontinuous Galerkin metods. 1. Introduction. An attractive feature of discontinuous Galerkin (DG) discretizations is teir suitability for a variety of different problem types as well as te flexibility regarding local mes refinement and even locally varying te order of te discretization. Wile initially te main focus as been on transport problems like yperbolic conservation laws interest as sifted recently towards diffusion problems. One simple reason is tat suc problems naturally enter te picture in more complex applications like te compressible or incompressible Navier-Stokes equations. By now numerous papers ave been devoted to well-posedness and stability issues as well as to te derivation of (a priori and a posteriori) error estimation [2, 1, 10, 12]. In comparison muc less seems to be known about efficient solution metods for te resulting systems of equations. An important step in tis direction as been [8] wic proposes a multigrid sceme (covering actually convection diffusion problems as well) tat does give rise to uniformly bounded condition numbers for diffusive problems provided tat (i) te underlying ierarcy of meses is quasi-uniform and (ii) te solution exibits a certain (weak) regularity. On te oter and, te local nature of te metod suggests a particular suitability for problems wit singular beavior calling for inomogeneous meses and possibly weak regularity requirements. Terefore te main objective of tis paper is to present a preconditioner tat is optimal in te sense of providing uniformly bounded condition numbers even for meses tat are sape regular but only locally quasiuniform and ave anging nodes, arising e.g. from local mes refinements. Moreover, no extra regularity assumptions are needed. Te approac is quite different from [8] and uses stable splittings and concepts from te teory of multilevel Scwarz scemes. As one important by-product of our analysis, we prove tat a simple splitting of any discontinuous, ig order piecewise polynomial space V := P k (T ) based on its first degree conforming subspace (te minimal conforming subspace) is stable wit respect to te standard mes dependent norm associated wit te DG discretization of an H 1 -elliptic boundary value problem (see (2.7)). Te essential (and necessary) requirement for tis is tat te triangulation T underlying V, tat may ave anging nodes due to local refinements, satisfies a certain grading property. A special and somewat simplified version of tese findings can be formulated as follows. Teorem 1.1. Let Φ := {φ i : i I } be a standard local basis of V, i.e. eac basis function is a piecewise polynomial supported in a single triangle, and let V c,1 := H1 0 (Ω) P 1 (T ) V denote te space Tis work as been supported in part by te Frenc-German PROCOPE contract 11418YB, by te European Commission Human Potential Programme under contract HPRN-CT , Breaking Complexity, by te SFB 401 and te Leibniz Programme funded by DFG. Institut für Geometrie und Praktisce Matematik, RWTH Aacen, Aacen, Germany, brix@igpm.rwt-aacen.de, ttp:// Institut de Recerce Matématique Avancée, CNRS - Université Louis Pasteur, Strasbourg, France, campos@mat.u-strasbg.fr, ttp:// Institut für Geometrie und Praktisce Matematik, RWTH Aacen, Aacen, Germany, damen@igpm.rwt-aacen.de, ttp:// 1

2 of continuous functions tat are piecewise affine on T. Ten c v 2 inf v c 2 v c V c,1, di R 1 + d i φ i 2 C v 2, v=v c + P i I i I d iφ i v V olds for constants c and C independent of T, provided tat te above mentioned grading property olds. Te layout of te paper is as follows. In Section 2 we set notation, formulate te model problem and recall some relevant facts about te symmetric interior penalty discontinuous Galerkin metod (SIPG). In Section 3 we briefly review te relevant aspects of additive Scwarz scemes tat provide te framework for our preconditioners. Specifically, in Section 3.1 we point out first tat a certain naive attempt motivated by wavelet-type preconditioners does not work. Tis is unfortunate since so-called multiwavelets would offer a simple tool in te present context. Te explanation wy tis fails is instructive in tat it igligts te necessity of working wit suitable splits of te trial spaces into a conforming part and a part tat could solely be controlled by te jumps across element edges. Section 4 is devoted to te identification of suitable stable splittings tat give rise to uniformly bounded condition numbers. Tere are essentially two crucial issues tat are peraps wort mentioning. First, te required grouping of stable splittings into conforming and nonconforming parts requires nontrivial conforming subspaces. Wen working wit local mes refinements wit anging vertices tis will be seen to boil down to certain mild grading conditions on te meses tat are discussed in Section 4.1. Second, given suc graded meses suitable decompositions into conforming and nonconforming subspaces can be obtained troug certain averaging projectors. Teir essential qualifying property, namely a certain Jackson-type estimate, is developed in Section 4.2. In fact, te main result of Section 4 is tat any split induced by suc an admissible projector into a conforming subspace of te wole trial space gives rise to optimal Scwarz preconditioners by simply taking te union of two stable splittings for te respective subspaces, see Teorem 1.1. Section 5 is ten devoted to te detailed concrete realization of a specific admissible averaging projector wose image is te space of continuous piecewise linear finite elements. In tis case stable splittings for te conforming part can be invoked from te literature wile te discontinuous piecewise polynomial basis of te wole trial space turns out to be a stable splitting for te corresponding nonconforming subspace. In tis case te conforming subspace is minimal. We empasize toug tat tis is only one possible realization among many oter ones. In Section 6 we present some first numerical experiments. We conclude wit indicating alternatives based on projectors onto maximal conforming subspace wose detailed analysis supported by more extensive numerical quantifications are deferred to a fortcoming paper. We sall sometimes write a < b to mean tat a can be bounded by some constant multiple of b were te constant does not depend on any parameters on wic a, b may depend. Likewise a b means tat bot a < b and b < a old. 2. Te Symmetric Interior Penalty Metod - A model problem. Altoug te subsequent developments carry over to iger spatial dimensions we sall confine te discussion for simplicity to te two-dimensional case. Tus, let Ω be a bounded polygonal domain in R 2 and let a(u, v) := (A u) v + buv (2.1) Ω were A is a uniformly positive definite 2 2 matrix and b is a nonnegative bounded function on Ω. To avoid quadrature issues, we sall always assume tat bot A and b are piecewise constant wit respect to a fixed coarse sape regular conforming triangulation T 0 = T 0 (Ω) of Ω. As usual tis means tat te ratio of te radii of te circumscribed and inscribed circles of all triangles remains uniformly bounded. It will be convenient to work wit closed triangles, i.e. te triangles K will always be assumed to be closed. We consider te following model problem: Given f L 2 (Ω), find u H 1 0 (Ω) (te usual Sobolev space of functions in L 2 (Ω) wose first order weak derivatives belong also to L 2 (Ω) and wose trace vanises on Ω), suc tat Ω a(u, v) = (f, v), v H 1 0 (Ω). (2.2) Te Lax-Milgram Teorem ensures te unique solvability of (2.2). In particular, we know tere exist positive constants c a, C a depending only on Ω, A, b suc tat for v 2 1 := v 2 L + 2(Ω) v 2 L 2(Ω) one as c a v 2 1 a(v, v) and a(v, w) C a v 1 w 1, v, w H 1 0 (Ω). (2.3) 2

3 We sall discretize (2.2) wit a discontinuous Galerkin sceme. To describe tis it will be convenient to employ te following localized inner products. For any domain G Ω we set (v, w) G := vw, a(v, w) G := (A u) v + buv. G 2.1. Te Symmetric Interior Penalty Metod on locally refined triangulations. Te DG discretization tat we sall describe next will be based on triangulations T = T (Ω) tat are generated from some initial sape regular conforming triangulation T 0 troug possibly local refinements. Te subscript terefore refers to a piecewise constant mes size function given by (K) := diam(k), K T. In principle, any kind of local refinement is permitted tat preserves sape regularity suc as partition based on newest vertex insertion or te standard partition of a triangle into four congruent subtriangles. One could tink of closing te partitions arising from suc local refinements to keep te triangulation conforming (wic means any two distinct closed triangles ave an empty intersection or sare a common face (edge or vertex)) wile preserving sape regularity. We stress toug tat nonconforming triangulations are permitted as well, i.e. anging vertices (sometimes called anging nodes) may occur, see Figure 2.1. Since te occurrence of anging vertices does complicate te construction of te preconditioner and its analysis somewat and since conforming sape regular triangulations may be viewed as simpler special cases we sall concentrate in te following on situations were anging vertices occur. As usual we require te triangulations to be graded meaning tat an edge may contain at most one anging vertex, see Figure 2.1. G G Fig Coarse T 0 (left), refinement (middle), and graded, sape regular T (rigt) Later we sall ave to impose a sligtly stronger grading property. Denoting by P k (G) te space of all polynomials of total degree at most k over te domain G, we associate wit any (admissible) triangulation T te trial space V := P k (T ) := K T P k (K) = {v = K T χ K P K : P K P k (K)} of all (possibly discontinuous) piecewise polynomials of degree at most k on T. Here χ K is te standard indicator function, i.e. χ K (x) = 1 for x K and χ K (x) = 0 else. Te subscript k is tus to be understood (in te same way as te mes size function ) as a degree function taking a constant integer value k = k(k) on eac triangle K T tat may vary for different K. However, we sall always assume tat a fixed upper bound k is never exceeded, i.e. k k for all K T. For te derivation of te symmetric interior penalty metod we refer e.g. to [1, 2]. To formulate it in te present setting we need te following notation. Let E be te set of edges of te triangulation T. Since anging vertices are permitted we sall adopt te convention tat wenever an edge as a anging node it is replaced in E by its two alves. Moreover, E,b, E,i denote te subsets of boundary and interior edges, respectively. Since triangles are always assumed to be closed sets, for any e E,i tere exist K, K T suc tat e = K K, and at least one of te two triangles as e as an edge. It will also be convenient to use te abbreviation v K := χ K v. Tus for any v V and K T it makes sense to denote by v K e te trace of te restriction of v to K on te subset e of K. As usual te jump of v across e is ten defined as [v] e := n K,e v K e + n K,e v K e, 3

4 were n K,e = n K,e denotes te unit normal vector on e pointing to te exterior of K. Wen e E,b is a boundary edge we tink of v being extended by zero outside Ω. Likewise averages on e are defined as { 1 {v} e := 2 (vk e + v K e ), e E,i, v K e, e E,b (and we sall drop te subscript e wen it is obvious from te context). Given T, consider te bilinear form a (v, w) := ( ) γ a(v, w) K { w} [v] + { v} [w] + [w] [v]. (2.4) K T e e e e E Here γ is a positive (possibly piecewise) constant tat as to be cosen appropriately to ensure stability of te metod, see relation (2.7) below. Its existence is usually argued under sligtly different assumptions on te underlying meses but te argument easily carries over. In fact, te only points to be addressed are peraps te occurrence of a anging vertex and a possible local variation of γ depending on te coefficients in (2.1). Applying a Caucy-Scwarz inequality to (2.4), one gets a (v, v) K T a(v, v) K 2 ( ) 1/2 ( e σ { nv} 2 L 2(e) e E e E e E ) 1/2 σ e [v] 2 L 2(e) + e E γ e [v] 2 L 2(e). Wen e = K K is an edge of K but only alf of an edge of K, as in Figure 4.1, we can write for v V te restriction v K as J C(K) vj. Hence, denoting by λ max / min (A) te maximal/minimal eigenvalue of A, and defining for ω Ω λ ω := inf x ω λ min(a(x)), te usual combination of a trace inequality wit an inverse inequality yields e { n v} 2 L 2(e) C( v 2 L 2(K ) + v 2 L 2(J) ) C λ ω := sup λ max (A(x)), (2.5) x ω max J =K,J λ 1 J a(v, v) K J, (2.6) were J K is te cild of K adjacent to K and te constant C depends on te sape properties of T 0 and k. Note tat writing for any θ (0, 1) v 2 L 2(K ) θ v 2 L 2(K ) + (1 θ)c22j(k ) v 2 L 2(K ) {, we could ave replaced te constant λ 1 J by min θ [0,1] max θλ 1 J, ) } C(1 θ)22j(j min x J b(x). Tis would allow us to get better estimates wen te diffusion coefficient degenerates and te zero order term is large. We sall dispense toug wit making tese distinctions in wat follows. At any rate, employing Young s inequality in combination wit (2.6) and a suitable coice of σ > 0 sows tat tere is indeed a positive γ suc tat a (v, v) 1 ( ) a(v, v) K + e 1 [v] 2 L 2 2(e) =: 1 2 v 2, v V, (2.7) K T e E wic yields te desired stability. Remark 2.1. If we allow γ to vary, we sould take γ σ 1/2 were for any edge of K, te σ and ence γ sould be of te order of λ 1 K up to constants depending on te sape properties of T 0 and te degree k of te piecewise polynomials, but not on. Te same type of estimates establises continuity a (v, w) C v w, v, w V, (2.8) for some uniform constant C tat could actually be made C = 2 wen taking γ sufficiently large but again of te order of λ 1 K locally. (In fact, continuity could be establised even in a larger infinite dimensional space, see e.g. [10].) Hence, te problem find u V suc tat a (u, v) = (f, v), v V, (2.9) as a unique solution, were a is te symmetric bilinear form defined on V by (2.4). Unfortunately, as in te case of conforming discretizations te corresponding linear systems of equations grow increasingly ill-conditioned wen te mes size decreases. For instance, for quasi-uniform T te spectral condition numbers of stiffness matrices are known to grow like 2, see e.g. [8]. 4

5 3. Additive Scwarz scemes. In tis section we explain wy a classical multi-wavelet approac fails to yield optimal preconditioners, and formulate te more general framework for additive Scwarz scemes in te DG context tat give rise to uniformly bounded condition numbers for te above type of discretizations under no additional regularity requirements on te solution u of (2.2) A way tat does not work. Scwarz type preconditioners for conforming discretizations are closely related to cange of basis preconditioners wen energy space stable multiscale bases are available [5, 6]. To explain tis, suppose tat Ψ = {ψ i : i I} is a basis of V and recall tat te spectral condition number κ(a Ψ ) of te stiffness matrix A Ψ := ( a (ψ i, ψ j ) ) is given by i,j I κ(a Ψ ) = ( sup v ) ( v T A Ψ v v T inf v v ) 1 v T A Ψ v v T. v Now according to (2.7), (2.8), writing v = (v i ) i I we ave for v = i I v iψ i, Tis immediately yields te following fact. Remark 3.1. If Ψ is stable in te sense tat v T A Ψ v v T v = a (v, v) v 2 v 2 l v 2. 2 l 2 v 2 i I v i 2 ψ i 2 (3.1) for any v = i I v iψ i in V, ten an optimal preconditioner is obtained by a symmetric diagonal scaling, namely we ave κ(da Ψ D) 1 wit D = diag ( ψ i 1 : i I). In te conforming case suc energy space stable bases are typically derived from L 2 -Riesz multilevel bases by rescaling. In te DG context it is very easy to construct L 2 -stable multilevel bases for V using te concept of multi-wavelets. One migt terefore be tempted to construct a cange-of-bases preconditioner by suitably rescaling suc a multi-wavelet basis. For suc a multilevel basis of V, a refinement T of te mes T entails expanding te current basis Ψ for V to te larger basis Ψ by adding a collection of new basis functions (on corresponding locally iger refinement levels) tat span a complementary space W of V so tat V = V W. In te case of multi-wavelets all basis functions would consist of globally discontinuous piecewise polynomials, and in te present context tis turns out to cause a principal problem tat we describe now. Consider a discontinuous basis function ψ i of Ψ, and assume tat it belongs to some space V. Clearly tere is at least one edge e in E for wic [ψ i ] L2(e) is positive. Now, if T is a refinement of T were e is uniformly refined into 2 l pieces, we ave ψ i 2 e E :e e 1 e [ψ i] 2 L 2(e ) = 1 2l e [ψ i] 2 L. 2(e) Hence ψ i tends to infinity as te edge e is furter refined. As a first consequence, tis sows tat te ψ i will not remain normalized wen progressing to iger levels of refinements causing te mes dependent norm to cange. Tis does not exclude yet stability in te sense of (3.1). On te oter and, te energy norm of a continuous v V reduces to a(v, v) 1/2 and ence does not depend on te mes. If ψ i remains uncanged wen expanding te basis of V to a basis of V, te associated coordinate v i does not cange eiter. Te desired stability of Ψ would tus yield a(v, v) = a (v, v) = v 2 v i 2 ψ i 2 v i 2 ψ i 2, i I 5

6 wic clearly implies tat v i = 0, since oterwise te (constant) quantity a(v, v) would ave to tend to infinity wit 0. From tis observation we infer tat wen Ψ is a uniformly stable multilevel basis for te spaces V, ten it must ave a subset Ψ c tat spans te conforming spaces V c := V H 1 0 (Ω). In particular, Ψ and V respectively decompose into Ψ = Ψ c Ψ nc and V = V c V nc in suc a way tat subsets of Ψ c and Ψ nc are stable bases for V c nc and V, respectively, and tat te splitting is stable wit respect to te discrete energy norm. Tis observation tat appears to exclude standard multi-wavelets will guide te subsequent developments Additive Scwarz preconditioners. It is of course not necessary to construct stable multilevel bases to produce efficient preconditioners for te problem (2.9) of finding u V suc tat a (u, v) = (f, v), v V. (3.2) In fact, te teory of additive Scwarz scemes offers a more flexible framework based on so-called stable splittings for te respective energy space wic in turn permit redundant systems, namely frames. Suc stable splittings for te DG setting will be constructed in Section 4.3 below. First let us briefly recall te notion of suc splittings and under wic circumstances tey give rise to good Scwarz preconditioners. Te collection S = { V i }i I, V i V, is called a ( -)stable splitting of V, uniformly wit respect to all triangulations T of te specified type, if i I V i = V and c S v 2 inf v i V i v= P i I v i { i I v i 2 } C S v 2, v V (3.3) olds wit constants c S, C S independent of T. Wit te aid of suc stable splittings one can transform equation (3.2) into an equivalent problem wic will be uniformly well conditioned. To tis end, te next ingredients are auxiliary inner products on te spaces V i, namely b i (, ) : V i V i R, i I, yielding norms tat are locally equivalent to, i.e. satisfy c b v i 2 b i(v i, v i ) C b v i 2, v i V i, i I, (3.4) for constants c b, C b depending at most on te maximal degree k, te sape properties of T 0 and possibly on te coefficients in te bilinear form a(, ) (see (4.37) below for concrete coices). Finally, define te operators P i : V V i and elements f i V i by b i (P i w, v i ) = a (w, v i ), b i (f i, v i ) = f, v i, v i V i, i I. (3.5) Note tat, wenever V i = span(θ i ) is a one-dimensional space, te application of P i just amounts to solving a linear equation wit a single unknown, namely P i w = d i θ i d i = a (w, θ i )/b i (θ i, θ i ). (3.6) Te central result we sall use reads ten as follows, see e.g. [14, 7]. 6

7 Teorem 3.1. Define P : V V and f V by Ten te problem (3.2) is equivalent to te operator equation P := P i, f := f i. (3.7) i I i I P u = f. (3.8) Moreover, if (3.3) and (3.4) old, ten te spectral condition number of te symmetric positive definite (wit respect to te inner product a (, )) operator P can be bounded by κ(p ) 2 C C bc S c b c S (3.9) tus by a constant independent of T, see (2.8), (3.3) and (3.4). Recall tat κ(p ) = λmax(p ) λ min(p ), were a (P v, v) λ max (P ) = sup v V a (v, v) v 0 and a (P v, v) λ min (P ) = inf v V a (v, v). v 0 Obviously te complexity of applying te operator P in some iterative procedure as linear complexity wen te dimension of te subspaces V i is uniformly bounded. Suc stable splittings will be constructed in Sections 4.3 and 5 below, see in particular Corollary 5.3 for one specific realization. Now, to interprete te above result let us view (3.2) as te operator equation L u = f, were L is defined by L w, v = a (w, v), w, v V. Moreover, defining L i : V i V i by b i (w i, v i ) = L i w i, v i (i.e. L 1 i is te approximate inverse of te restriction of L to V i ), and denoting by Q i : V V i te V i -ortogonal projection, we ave by definition, Terefore, we can write as usual ( P u = i I L 1 P i = L 1 i Q i L. ) ( i Q i L u and f = i I L 1 i Q i )f, (3.10) i.e. te operator C := i I L 1 i Q i (3.11) can be viewed as a preconditioner. 4. Stable splittings for te DG metod. Tis section is devoted to te construction and analysis of stable multilevel splittings for te DG metod tat can be used in Teorem 3.1. Tere are two ingredients tat are peraps wort empasizing. Te first one is tat stable splittings for V can simply be assembled from stable splittings for an appropriate conforming and nonconforming part of V, respectively. Te appropriateness will inge on certain admissible averaging operators tat are required to satisfy a Jackson estimate. Second, te locality of suc an estimate, owever, turns out to impose a sligtly stronger notion of gradedness (see Definition 4.1 below) for te underlying meses (witout actually inflating te complexity, see Remark 4.2 below) wic we sall terefore address first. 7

8 4.1. Admissible triangulations extended grading. We ave already defined a standard grading requirement proibiting an edge to ave more tan one anging vertex. Te grading tat we sall ave to impose on te triangulations will actually be sligtly stronger, but essentially keeps te complexity equivalent, see Remark 4.2 below. To limit tecnicalities we sall only consider triangulations tat are obtained by local subdivisions into four congruent subtriangles, starting from a conforming and sape regular coarse T 0, as in Figure 2.1. As a consequence, every element of T as a refinement level j(k), j(k) = 0 meaning tat K T 0. From te sape regularity of T (inerited from tat of T 0 ), one ten easily infers K := meas(k) ( K ) 2 2 2j(K) for any K T (4.1) wit constants tat depend only on te coarse T 0. Definition 4.1 (graded triangulations). A triangulation T obtained from refining a conforming coarse T 0 as described above is called graded, if every anging vertex n is te midpoint of an edge spanned by two vertices, denoted by n and n in Figure 4.1, left, and if in suc a case, n and n are never anging vertices temselves. Note tat tis grading requirement implies te usual one, i.e. tere is at most one anging vertex per edge. K n n K n K Fig For T be graded (te rigt one is not), n and n must be regular vertices. Remark 4.1 (necessity of te (extended) grading). Te above grading requirement is necessary for Teorem 1.1 (and furter similar splittings) to old. To see te validity of te above claim consider one triangulation T tat as been obtained from one single coarse triangle T 0 = {K 0 } by recursively refining te central cild at eac level up to some j, as sown in Figure 4.1, rigt. Clearly, suc a T satisfies te classical grading requirement permitting only one anging vertex per edge, but not te above stronger one. In particular, all its vertices except tose on Ω are anging, so tat H0 1 (Ω) P 1 (T ) only contains 0. Hence Teorem 1.1 would imply ere tat for any basis {φ i : i I } of V := P k (T ) consisting of local polynomial pieces normalized in L, one as v 2 d i 2 for every v = d i φ i V, (4.2) i I i I wit constants independent of te refinement level j. Consider now te piecewise constant function v = χ K, were K is te central triangle on level two. If te φ i are interpolatory, like in a local Lagrange basis, one as d i = 1 for all but finitely many indices i and te rigt and side in (4.2) grows proportionally to j. On te oter and, since v is continuous inside K, it can be seen from te definition of te energy norm, given in Section 2.1, see (2.7), tat v doesn t depend on te furter refinements of T, wic clearly contradicts (4.2). We sall next give an algoritmic caracterization of te grading property in Definition 4.1. Since te triangulations T of interest result from (possibly local) refinements, suc a refinement istory defines a tree T, were T is te set of leaves (i.e. final nodes) of T. We denote by P (K) and C(K) te parent and set of cildren, respectively, of a given triangle K. Tis means any cild K C(K) is obtained by subdividing its parent K = P (K ) once, according to te given rule. By P 2 (K) = P (P (K)) we denote te grandfater of K wen j(k) 2. 8

9 Also, by T j we denote te jt uniform dyadic refinement of T 0. Clearly, for any level j tis is a partition of Ω. Given any triangle K in T, we ten let its neigbors N(K) be defined as te collection of triangles at level j(k), i.e. belonging to T j(k) (but not necessarily to T ) tat sare an edge wit K. Finally, N 2 (K) := K N(K)N(K ) consists of te neigbors of te neigbors of K. In tese terms a constructive formulation of te grading property reads as follows. Remark 4.2 (algoritmic formulation of te grading requirement). A triangle K in T is said to ave te G-property if it is of level j(k) 1 or if - in te case tat P (K) is not te central cild of P 2 (K), every triangle of N(P (K)) is in te tree T, or - in te case tat P (K) is te central cild of P 2 (K), every triangle of N 2 (P (K)) is in te tree T. Ten te following properties old, wose proofs are left to te reader. 1. A triangulation T is graded in te sense of Definition 4.1 if and only if all its elements satisfy te G-property. 2. Given any multilevel triangulation T, its smallest graded refinement is built by imposing recursively (starting from te igest levels) te G-property on all its elements. Terefore te G-property is a practical tool for constructing graded triangulations. Moreover, it can be cecked tat te set T := T K b T C(N 2 (P (K))) (4.3) is indeed a tree, and tat te corresponding set of leaves T is a refinement of T in wic any triangle as te G-property. T is ten a graded refinement of T, and it is a standard exercise (see for instance Lemma 2.4 in [4]) to derive from (4.3) te following complexity estimates #(T ) #( T ) #( T ) #(T ) wic olds wit uniform constants. Tus, imposing our (extended) grading keeps te complexity of adaptive triangulations up to uniform constants. We sall encefort assume tat T is graded in te sense of Definition Admissible conforming subspaces. Given a graded triangulation T, it remains to construct energy stable splittings to be used in Teorem 3.1. Motivated by te observations made in Section 3.1, we sall first look decomposing V into a conforming and nonconforming part V c nc and V, respectively. However, it turns out tat in te more general framework offered by Scwarz preconditioners, even te meaning of V c can be relaxed in tat it need not exaust all of V H0 1 (Ω). Different conforming subspaces will be seen to lead to different versions of preconditioners tat are all asymptotically optimal in te sense of uniformly bounded condition numbers. Wat matters is tat te conforming part is te range of a suitable averaging projector A into some subspace of V H0 1 (Ω). In tis section we formulate te key property tat qualifies te averaging projector as suitable, namely a certain local Jackson-type estimate involving for eac K T te localized energy norms v 2,ω := a(v, v) K + e 1 [v] 2 L 2(e) K T :K ω e E :e ω for certain neigboroods ω = ω(k) Ω of any K T. To define ω(k) we need some furter preparations and notational conventions. Te set of vertices viz. first degree nodes will be denoted by N 1 (K) := {vertices of K} and N,1 := N 1 (K). In connection wit iger order elements we sall ave to consider later corresponding sets of iger kt degree nodes tat will be denoted by N k (K), see Section K T

10 Moreover, it will be convenient to set for any closed domain D T (D) := {K T : K D }. Similarly we define te sets N,1 (D) and E (D) for mes elements toucing D. For instance, T (n) consists of te triangles tat sare te vertex n (including anging vertices). Note tat wen an edge of a triangle K contains a anging vertex te collections N 1 (K) and N,1 (K) differ by tat anging vertex. Unfortunately, due to anging vertices, straigtforward neigboroods based on tese notions do not suffice and te construction of te domains ω(k) appearing in te announced Jackson estimate will actually rely on some extended sets of mes quantities. To tis end, recall tat for any vertex n tere are two possibilities: eiter it is anging, i.e. tere is one triangle K in T (n) for wic n is not a vertex, and ten n is te midpoint of one edge e = [n, n ] of K, see Figure 4.1, or it is not. Wit tese notations we first define { N,1(n) {n, n, n } if n is anging, := (4.4) {n} oterwise, i.e. if n is regular. Recall tat te grading we impose on T precisely means tat bot n and n are regular, see Definition 4.1. For any triangle K T we ten set N,1(K) := N,1(n) N,1, (4.5) E (K) := n N 1(K) n N,1 (K) E (n) E (4.6) and and finally define te domain T (K) := ω(k) := n N,1 (K) T (n) T, (4.7) K T (K) K Ω (4.8) as te union of triangles tat are in contact wit te extended set of vertices N,1 (K). An illustration is presented in Figure 4.2, were te sets N,1 (K), E (K) and T (K) are represented by wite vertices, bold edges and gray triangles, respectively. Note in particular tat te above definitions yield a(v, v) K + e 1 [v] 2 L 2(e) v 2,ω(K). (4.9) K T (K) e E (K) We are now prepared to formulate te key property of averaging projectors A tat will be crucial for te subsequent developments. Property 4.3. A linear projector A tat takes V into V H 1 0 (Ω) is called admissible if for any v V one as te following Jackson estimate (I A)v L2(K) C 2 j(k) v,ω(k), K T, (4.10) wit a constant C tat may depend on te sape properties of T 0 and on k (and on concrete realizations of γ). Note tat te domains ω(k) defined by (4.8) are local (closed) neigboroods of te triangles K T satisfying a bounded overlapping property, i.e. sup # ( {K T : ω(k) ω(k ) } ) 1 (4.11) K T 10

11 K Fig Te neigborood ω(k) consists of te union of te (gray) triangles in T (K). olds wit a constant tat depends only on T 0. We postpone te concrete construction of admissible A and te verification of Property 4.3 to Section 4 and discuss first te consequences regarding stable splittings. Te significance of Property 4.3 lies in te following two consequences, summarized in Propositions 4.2 and 4.3. Te first one concerns te energy stability of an admissible A. Proposition 4.2. Assume tat A satisfies te Jackson estimate (4.10). Ten one as (I A)v < v, Av < v, v V, (4.12) were te constants depend on k, te sape properties of T 0 as well as on C and C a defined in (4.14) below. As a consequence we ave v Av + (I A)v C A v, (4.13) wit te above dependence of te constant C A in te upper bound. Proof: Since te jumps of (I A)v equal te jumps of v we need only estimate K T a((i A)v, (I A)v) K. To keep track of te dependence of te various constants, recall tat λ K is defined by (2.5) and let C a,k := 2 max {λ K, 2 2j(K) b L (K)} and C a := max K T C a,k < C a, (4.14) see (2.3). Since (I A)v is a polynomial on K, a standard inverse estimate yields a((i A)v, (I A)v) K cc a,k 2 2j(K) (I A)v 2 L 2(K) < C C a,k v 2,ω(K), were c depends on k and T 0 and were we ave used (4.10) again in te last step. On account of te bounded overlap of te ω(k), K T (see (4.11)), and since Av v + (I A)v, tis concludes te proof. Tus, defining V c := AV and V nc := (I A)V, (4.15) as possible candidates for conforming and nonconforming parts of V, te above observation (4.13) already indicates tat a stable splitting for all of V of te form (3.3) can be composed of stable splittings for te individual subspaces V c nc and V, respectively. In fact, tis would allows us to invoke known splittings for conforming spaces, see e.g. [3, 5, 7, 14] and Section Now, te second consequence of Property 4.3 is a localization of V nc tat will simplify te identification of stable splittings for te nonconforming part. In wat follows we sall make frequent use of te 11

12 following simple consequences of standard trace inequalities, rescaling arguments and te fact tat all norms on a fixed finite dimensional space are equivalent. Remark 4.4. For any polynomial P P k, any K T and any edge e of K, te following relations old. (i) One as P L (e) 2 j(k)/2 P L2(e), (4.16) were te constants depend only on te degree k of P and on T 0. (ii) Tere exists a constant c depending only on te degree k of P and on T 0 suc tat χ K P c(c a,k + 1) 1 2 { 2 j(k) P L2(K), P L (K), (4.17) were te constant c depends only k and te sape properties of T 0 and C a,k is given by (4.14). Observe ten tat splittings for V nc will indeed take a simple form due to localization. Proposition 4.3. Assume tat A is admissible, i.e. satisfies (4.10). Ten tere exists a constant C 0, depending only on k, te sape properties of T 0, on C a defined in (4.14), and on te constant C in (4.10) suc tat for w K := χ K w w 2 w K 2 C 0 w 2, w V nc := (I A)V. (4.18) K T Proof: Te lower inequality in (4.18) is due to te fact tat for any edge e K K one as [w] 2 L 2(e) wk 2 L 2(e) + wk 2 L 2(e). As for te upper inequality, since w V nc and wk is a polynomial on K, we infer from (4.17) and (4.10) (noticing (I A)w = w) tat w K 2 < (C a,k + 1)2 2j(K) w K 2 L 2(K) < (C a,k + 1)C w 2 (ω(k)), (4.19) were C a,k can be bounded by C a from (2.3). Te upper inequality follows now again from (4.11) Energy stable splittings for V. As mentioned above, te stability of A asserted by Proposition 4.2 implies tat stable splittings for te individual parts V c nc and V give rise to stable splittings for all of V. Moreover, te localization in V nc given by Proposition 4.3 togeter wit te fact tat te frame for te nonconforming part need actually not be contained in V nc greatly simplifies finding suc a complementary frame. Teorem 4.4. Let {Vi c} i I c be a stable splitting for V c, i.e. c 1 a(v, v) inf a(v i, v i ) C 1a(v, v), v V c, (4.20) v i V c i v= P i I c vi i I c olds for some constants c 1, C 1 independent of T. Ten one as for any v V c S v 2 inf v v K, v K 2 i + a(v i, v i ) C S v 2, (4.21) K T v= P K T v K + P i I c vi were te v K and v i belong to χ K P k (K) and V c i, respectively, and c S = (max{2, c 1 1 }) 1, C S = C A max{c 0, C 1 }. Tus te collection i I c S = {χ K P k (K) : K T } {V c i : i I c } (4.22) 12

13 is a stable splitting for V in te sense of (3.3). Proof: By definition of te infimum we ave { inf v v K, v K 2 i v= P K T v K + P + } a(v i, v i ) i I c vi K T i I c { } inf v v K 2 K (I A)v= P K T v K K T { } + inf a(v v i, v i ) i Av= P i I c vi i I c C 0 (I A)v 2 + C 1 Av 2 C S v 2, (4.23) were we ave used Proposition 4.3 and (4.20) in te second step (remember tat a(, ) and 2 coincide on V c ) and Proposition 4.2 in te last step. Tus C S = C A max{c 0, C 1 }. Tis confirms te upper bound in (4.21). In order to confirm te oter direction consider any expansion wit ṽ K χ K P k (K) and ṽ i Vi c. Writing first we note tat by (4.20) v 2 2 = a(v 2, v 2 ) c 1 1 inf v i v = ṽ K + ṽ i =: v 1 + v 2 K T i I c v 2 2( v v 2 2 ), (4.24) v 2= P i I c vi wile as in te proof of Proposition 4.3 one as for any v 1 = K T ṽk V v 1 2 Tus, we can combine (4.25) and (4.26) to conclude tat 1 2 a(v i, v i ) c 1 1 a(ṽ i, ṽ i ), (4.25) i I c ( v 2 2 ṽ K 2 + c 1 1 K T i I c K T ṽ K 2. (4.26) i I c ) a(ṽ i, ṽ i ). Since te decomposition of v was arbitrary we ave sown te lower inequality in (4.21) for c S = (max{1, c 1 1 }) 1. Teorem 4.4 covers various possible specifications of splittings wic we sall begin to discuss now. Recall first tat, in principle, energy stable splittings for conforming finite element spaces are well understood. Of course, a concrete identification of collections {Vi c} i I c depends on te concrete realization of an admissible projector A, i.e. on te specific subspace AV. We postpone tis issue to Section 5. At tis point we note first tat te subspaces VK nc := χ KP k (K) are of uniformly bounded finite dimension depending only on k and terefore could be furter broken down. In fact, any (reasonable) basis for V will be seen to be an energy stable frame for V nc. To describe suc a basis it is convenient to introduce te canonical k-mes N k (K) := { p = p β := 1 k 2 β j n j : β Z 3 +, j= j=0 } β j = k, (4.27)

14 n 2 β = (1, 1, 1) n 0 β = (0, 3, 0) β = (2, 1, 0) n 1 Fig Canonical mes for k = 3, N 3 (K). induced on te triangle K, see Figure 4.3. Of course, for k = 1 we simply ave N 1 (K) = {n 0,..., n 2 }, te set of vertices of K itself, as used before. Note tat te β/k are te barycentric coordinates of te mes points p = p β, a fact tat will be used later again. Clearly we ave dim(p k ) = #N k (K). Now let {P p : p N k ( ˆK)} be a fixed basis for P k, were ˆK is te standard triangle wit vertices (0, 0), (1, 0), (0, 1), and define for K = F ˆK T, F = F K affine, φ K,p := χ K P p F 1 K, p N k(k). (4.28) Obviously, Φ := {φ K,p : p N k (K), K T } is a basis for V. Later, te reference polynomials P p will be (essentially) normalized in L (or in H 1, wic is equivalent in R 2 ). Tus one as P p L ( ˆK) 1, {a p} p Nk ( ˆK) L ( l a p P p ˆK), (4.29) p N k ( ˆK) wit constants depending only on k (and te specific coice of te polynomial basis). Teorem 4.4 yields ten te following general result, were it is seen tat a complementary frame is simply given by a union of local polynomial basis functions tat do not need to belong to te nonconforming part V nc but only to span it. Corollary 4.5. Assume tat A is admissible (i.e. satisfies te Jackson estimate (4.10)) and let {Vi c : i I c} be any energy stable splitting for V c := AV. Moreover, let te basis functions φ K,p be constructed as above, i.e. tey satisfy (4.28), and assume tat te set of indices satisfies Ten, setting V nc i V nc I nc {i = (K, p) : p N k (K), K T } (4.30) := (I A)V span{φ i : i I nc } =: span Φ nc. (4.31) := span φ i, te collection S = {V nc i is an energy stable splitting for V in te sense of (3.3). : i I nc } {V c i : i I c } (4.32) Proof: In view of Teorem 4.4, it suffices to estimate te local contributions w K, w K = χ K w, for w V nc. More precisely, we sall prove tat K T w K 2 < (K,p) I nc w K,p φ K,p 2 14 < K T w K 2 (4.33)

15 olds wit constants depending only on k, te sape properties of T 0, and te constants in (4.10), (2.3). Indeed writing w K = w K,p φ K,p, (K,p) I nc we note from (4.28) and (4.29) tat φ K,p L (K) 1. Hence, by (4.17) we ave w K,p φ K,p 2 < (C a,k+ 1) w K,p 2 wit a furter constant factor depending on k and te sape properties of T 0. Using te fact tat only a fixed finite number (depending on k) of basis functions are supported on eac K, and taking te normalization (4.29) into account, we conclude tat w K 2 wk p 2 < < (K,p) I nc (K,p) I nc w K,p φ K,p 2 w K,p φ K,p 2 < { w K,p φ K,p }(K,p) I nc L (K) 2 l = w 2 L (K), (4.34) were te constants depend only on k as well as on C a,k, γ in (4.17) and (4.29). Now we use again a scaling/norm equivalence argument and (4.10) to obtain w L (K) < 2 j(k) w L2(K) < C w (ω(k)), (4.35) were te remaining unspecified constant factor depends only on k, and te sape properties of T 0. Finally, we note again tat for any edge e K K one as [w] 2 L 2(e) 2 wk L + 2(e) 2 wk L, 2(e) ence w 2 (ω(k)) w K 2. (4.36) K T :K ω(k) Combining now (4.34), (4.35), (4.36) and using again (4.11), yields (4.33) and completes te proof. For te concrete realization of te preconditioned problem (3.8), it now remains to identify viable coices of te auxiliary bilinear forms b i (, ) for i I := I c Inc. Here Ic, Inc are te index sets for stable splittings {Vi c nc }, {Vi } of V c, V nc, respectively, as discussed above. Likewise, te conforming splitting typically contains te full coarse grid space. In general, let ω(i) denote te union of all te supports of te nodal basis functions spanning te subspace Vi c (wic could be all of Ω wen i refers to a full coarse level space). Some admissible coices for te b i (, ) are te following: a(v, w) K + e E,e K e 1 e [v][w] or K 1 (v, w) K + e E,e K e 1 e [v][w] wen i = (K, p) Inc (4.37) a(v, w) ω(i) or ω(i) 1 (v, w) ω(i) wen i I c. Remark 4.5. In all te above cases in (4.37) tere exist constants c b, C b depending only on te degree k, te sape properties of T 0 and possibly on te coefficients in te bilinear form a(, ), suc tat (3.4) olds. Tus we can summarize tese findings as follows. Teorem 4.6. Assume tat A is admissible. Ten te additive Scwarz preconditioner based on te te above splittings (for te respective realizations of V c, see Section 5.1.1) and for any of te bilinear forms from (4.37) is asymptotically optimal in te sense of Teorem 3.1. Te robustness of te sceme wit respect to te diffusion coefficients in a(, ) will be discussed in more detail elsewere. 15

16 5. Projecting onto te minimal conforming subspace. In principle te results of te previous section offer a general framework for a family of stable splittings and associated preconditioners. In tis paper we are content wit one specific realization of an admissible projector A and te identification of corresponding stable splittings for te conforming part. Specifically, we coose A = A 1 so as to map V into continuous piecewise affine elements - in tis sense te smallest possible subspace of H 1 0 -conforming trial functions, i.e. V c,1 := A 1 V = H 1 0 (Ω) P 1 (T ). (5.1) In general it differs from V H 1 0 (Ω) wenever iger degrees are permitted. An alternative would be to ave A = A k produce globally continuous finite elements of te same degree as te original discontinuous elements and in tat sense to split off te maximal conforming subspace in V, i.e. V c,k := A k V = V H 1 0 (Ω). (5.2) Tis will be elaborated on in a fortcoming paper. In principle, suc averaging operators are familiar in oter contexts concerning DG discretizations, see e.g. [10]. Te form of suc projectors in te presence of anging vertices is to our knowledge less familiar. Since anging vertices do appear to add some complication we sall give in wat follows an explicit selfcontained construction Te construction of A 1. Te construction of A 1 is inspired by te results in [10, 11] but as to cope wit anging nodes. For tis reason, let us remember tat we call regular tose vertices tat are not anging. In particular, every vertex on te boundary Ω is regular. Since boundary vertices are subject to a zero boundary condition in H 1 0 (Ω), we can define A 1 v by prescribing its nodal values at every interior regular vertex n, following (A 1 v)(n) := 1 # ( T (n) ) K T (n) Since V c,1 is always a subset of V (tat is, for any degree distribution k), te space stands as a possible candidate for complementing V,1 c, i.e. we ave v K (n). (5.3) V nc,1 := (I A 1 )V (5.4) V = V c,1 V nc, Energy stable splittings for V,1 c. To describe an energy frame Φc,1 for V,1 c we can essentially resort to known results. In fact Φ c,1 as multilevel structure and is comprised by all nodal continuous basis functions associated wit all predecessors of T witin te tree T. To describe tis, let us denote again by T j te jt uniform dyadic refinement of T 0 and let T j := {K T j : K T s.t. K K} = T j T, (5.5) i.e. T j is comprised of all level j triangles tat appear in te refinement istory leading to T. Moreover, let Ω j := {K : K T j } te area covered by te elements in T j. Te subspace V,1 c := A 1V consists of all continuous piecewise linear functions on T. Consider now N j,c j,1 te set of tose vertices tat belong to a triangle in T and lie in te interior of Ω j, i.e. N j,c,1 := {N 1 (K) \ Ω j : K T j }. 16

17 Note tat since anging vertices are always located on some boundary Ω j j,c, te vertices in N neiter anging, nor tey lie in te boundary of te corresponding domains. Now define te multilevel index set,1 are I,1 c := {i = (j, n) : j = 0,..., j, n N j,c,1 }, (5.6) were j is te maximal level appearing in T. Wit eac i = (j, n) I,1 c we associate now te standard nodal (piecewise affine) at function ϕ c j i at te vertex n supported on te star of triangles in T saring n. Clearly ϕ c i is continuous, and it immediately follows from (4.17) tat ϕ c i = a(ϕ c i, ϕ c i) 1/2 1, i I c,1, (5.7) olds wit constants depending only on te constants in (2.3) and on te sape properties of T 0. Now set Φ c,1 := {ϕ c i : i I c,1}. It is easy to see tat Φ c,1 spans V,1 c and it is known (see e.g. [14, 7, 5]) tat tis collection is an energy-stable splitting, i.e. a(v, v) inf v v i R v= P i 2 a(ϕ c i, ϕ c i), v V,1, c (5.8) i I,1 c viϕc i I c i,1 were te constants depend only on T 0 and te constants c a, C a Te Jackson estimate for A 1. We sall confirm now te validity of Property 4.3 for A = A 1. To tis end, let us first establis te following intermediate result tat is only valid for piecewise affine functions. Remember tat te different sets E (n), N,1 (n), T (n),... wic denote local neigboroods made of mes elements, are defined in Section 4.2. Lemma 5.1. If v P 1 (T ) is a piecewise affine function, ten we ave for any node n N,1 (v A1 v) K (n) [v] L (e), K T. (5.9) n N,1 (n) e E (n ) Proof: We begin wit te following basic estimate, independent of te operator A 1 : at any vertex n N,1, we ave (v K v K )(n) [v] L (e), K, K T (n) (5.10) e E (n) for any v V. Indeed, since K and K do not necessarily sare an edge, write T (n) = {K 1,..., K M } te (closed) triangles containing n, in suc a way tat K i and K i+1 ave te edge e i E (n) as teir intersection, and assume tat K = K M and K = K m wit m < M (observe from te definition of te set E tat tis is possible even wen n is a anging vertex). It is ten easily seen tat M 1 M 1 (v K v K )(n) (v Ki+1 v Ki )(n) [v] L (e i), i=m wic clearly gives (5.10). Actually, because te jumps over boundary edges are defined as if v was extended to 0 outside Ω, tis argument yields te following estimate wen n is on Ω v K (n) [v] L (e), n Ω, K T (n). (5.11) e E (n) 17 i=m

18 Now to prove te assertion of te lemma we consider n K and distinguis tree cases: (i) n is on te boundary Ω, (ii) n is a regular vertex in te interior of Ω, and finally (iii) n is anging, i.e. tere is (only) one triangle tat contains n and for wic n is not a vertex. Note tat in te two first cases we ave N,1 (n) = {n}. In te case (i), (5.11) readily yields Lemma 5.1 since ten we ave A 1 v = 0. In te case (ii) were n is interior and regular, by definition (5.3) of A 1, we ave (v A 1 v) K (n) 1 #(T (n)) K T (n) (v K v K )(n), (5.12) so tat Lemma 5.1 now follows from (5.10). Finally, in te case (iii) were n is anging, denote by K te unique triangle in T (n) for wic n is not a vertex, and by [n, n ] te edge of K for wic n is te midpoint. Since bot v and A 1 v are affine on K, we ave (v A 1 v) K (n) 1 ( (v A 1 v) K (n ) + (v A1 v) K (n ) ). (5.13) 2 Now from te grading of T, we know tat n and n are never anging temselves, so tat (5.12) applies, providing, on account of (5.10), (v A1 v) K (ň) e E (ň) [v] L (e), ň = n or n. (5.14) Writing ten (v A1 v) K (n) v K (n) v K (n) + (v A1 v) K (n), we prove (5.9) in te case (iii) by combining te estimates (5.13), (5.10) and (5.14), since ten we ave by definition N,1 (n) = {n, n, n }, see (4.4). We are now ready to verify te validity of Property 4.3 for A 1. Proposition 5.2. Te projector A 1 as Property 4.3 were C in (4.10) depends only on te sape properties of T 0, on k, and on max K ω(k) λ 1 K. Proof: To estimate te projection error (I A 1 )v consider te interpolation operator P 1 : V P 1 (T ) given by for wic te estimate {P 1 v} K (n) := v K (n), K T, n N 1 (K), (5.15) (I P 1 )v L2(K) 2 j(k) v L2(K) (5.16) is well known wit a constant depending only on te sape properties of T 0. In view of (5.3), it is readily seen tat A 1 P 1 = A 1, so tat te projection error reads Writing w := (I A 1 )P 1 v P 1 (T ) it follows tat (I A 1 )v = (I P 1 )v + (I A 1 )P 1 v. (I A 1 )v L2(K) 2 j(k) v L2(K) + w L2(K). (5.17) Now, denote by {φ 1 (K,n) : n N 1(K)} te discontinuous nodal basis of P 1 (K). On any triangle K T we ave w L2(K) w K (n) φ 1 (K,n) L 2 2 j(k) w K (n). n N 1(K) 18 n N 1(K)

19 Applying ten Lemma 5.1 to P 1 v, and recalling tat te set E (K) is given by (4.6), we find w L2(K) 2 j(k) [P 1 v] L (e), e E (K) were te constant depends only on te sape properties of T 0. Now assume for te moment tat for any edge e E tere is one triangle K e for wic e K e, and suc tat te inequality [P 1 v] L (e) [v] L (e) + v L2(K e) (5.18) olds wit a constant tat only depends on te sape properties of T 0. Since for every e in E (K), any suc K e is in T (K), see (4.6) and (4.7), tis would yield w L2(K) 2 j(k) [v] L (e) + v L2(K ). e E (K) K T (K) Now since [v] is a polynomial on any edge e E, we recall from (4.17) tat [v] L (e) e 1/2 [v] L2(e). Moreover, as in (2.6) eac term v L2(K ) can be bounded by Cλ 1 K a(v, v) K were C depends only on k and te sape properties of T 0. Note tat tis means tat, aside from te dependence on k and te sape properties of T 0, te constant is of te order C max γ e, (5.19) e ω(k) see te remarks following (2.7). Since te sets E (K) are of uniformly bounded cardinality (depending only on T 0 ), we find (upon using also (5.17) and (4.9)) tat (I A 1 )v L2(K) C 2 j(k) v (ω(k)), were te constant C is of te form C max K ω(k) λ 1 K, C depending only on k and te sape properties of T 0. Tis completes te proof of Lemma 4.3. Tus, it only remains to prove (5.18). Here tree situations may occur: eiter (1) te edge e is on te boundary Ω, or tis is not te case and we can write e = K e K e wit K e, K e T. Now eiter (2) e is an edge of bot K e and K e, or again (3) tis is not not te case. In te two first cases, we observe tat [P 1 v] e = P 1 ([v] e ), so tat [P 1 v] L (e) [v] L (e) is obvious. Now according to te structure of graded triangulations, te only possible configuration in case (3) is represented in Figure 4.1 wit e = [n, n]. Setting ten K e and K e to be te coarser and finer triangles, respectively (i.e. K and K in Figure 4.1), we observe tat P 1 v is affine on K e. Terefore Writing { [P 1 v] L (e) = max v Ke (n ) v K e (n ), 1 ( v K e (n ) + v Ke (n ) ) } v K e (n). 2 1 v K e (n 2( ) + v Ke (n ) ) = 2( 1 v K e (n ) v Ke (n) ) + 1 ( v K e (n) v Ke (n ) ) + v Ke (n), 2 we find (wit an absolute constant) [P 1 v] L (e) [v] L (e) + v Ke (n ) v Ke (n) + v Ke (n) v Ke (n ). Using a scaling argument togeter wit te fact tat v is polynomial on K e, one finally verifies tat v Ke (n) v Ke (ň) v L2(K e), ň = n or n, 19