Adaptive Finite Element Method

Transcription

1 39 Capter 3 Adaptive inite Element Metod 31 Introduction As already pointed out in capter 2, singularities occur in interface problems Wen discretizing te problem (221) wit inite Elements, te singularities impair te approximation properties of inite Elements o reduce te approximation error we will use an a-posteriori approac relying on a-posteriori error estimates is approac as te advantage, tat it allows for te treatment of te singularities in two and tree space dimensions and tat it works witout knowledge of parameters depending on te singularities ese estimates are reliable and efficient and for a large class of problems also robust Robustness means, tat variations of te diffusion coefficient, ie te amount of te jump discontinuity, does not enter in te error bounds e adaptive procedure consists in refining te mes on te basis of te a-posteriori error estimators Numerical experiments sow convergence rates measured in terms of nodes of te inite Element approximation sceme of te same order as for regular problems or te derivation of te error estimators we will use certain robust interpolation properties of inite Element spaces erefore, it is necessary to restrict to te class of quasi-monotone diffusion coefficients [62] [24] In tis class we prove robustness of te a-posteriori error estimators Recall from capter 2, tat quasi-monotone diffusion coefficients guarantee regularity H 1+1/4 independent of te bounds of te diffusion coefficient k e outline of tis capter is as follows e problem setting and te notation used are introduced in section 32 We discuss approximation properties of inite Elements on uniform grids (section 33) and on adapted grids (section 34) o prepare te proof for te upper bound of te error by te error estimator, we need interpolation results (section 35) In section 352 we extend te definition of te quasimonotonicity given in te capter 2 to te 3D case A robust interpolation operator, wic is a sligt modification of tat from [24], will be defined in section 353 We address te open question about connections between regularity properties and interpolation properties in section 354 e main results of tis capter are presented in section 36 We introduce residual based error estimators in section 362 urter we discuss estimators wic are based on te solution of local problems and propose a new estimator in section 363

2 40 32 PROBLEM SEING We also discuss an approac wic relies on ierarcical bases in section 371 and a Zienkiewicz-Zu type estimator in section 372 Extensions to problems wit a mass term and Caucy boundary conditions are given in section 38 e application of te a-posteriori tecniques to a parabolic problem is demonstrated in section Problem setting 321 Approximation wit inite Elements e continuous problem is essentially problem 322 of capter 2, but as we allow for non-omogeneous boundary conditions we repeat te definition of te continuous problem Let Ω be a Lipscitz domain in R d,d =2, 3 wit polygonal (polyedral) boundary e domain Ω can be partitioned into a finite sum of subdomains Ω i wit polygonal (polyedral) boundary As before tere is a function k being constant on eac subdomain and fulfilling te bounds (321) δ k(x) δ 1 x Ω for some δ>0 In distinction to section 22 we now do not demand tat te subdomains Ω i are Lipscitz Let te boundary Ω be decomposed into Ω =Γ D Γ N,meas d 1 (Γ D Γ N )=0and meas d 1 (Γ D ) > 0 Let g D H 1/2 (Γ D ) be given ere is an extension of g D onto a function defined in H 1 (Ω) and aving g D as trace on Γ D Let us denote tis extension also by g D Let g N L 2 (Γ N ) and f L 2 (Ω) be given We define te space V := { u H 1 (Ω) : u ΓD =0 } e variational form of te problem is as follows: seek u g D + V satisfying: (322) Ω k(x) u(x) v(x) dx = Ω f(x) v(x) dx + g N (x) v(x) dx Γ N v V We introduce a discrete problem troug inite Element spaces V V wit continuous and piecewise linear functions e underlying triangulation is refered by We assume tat te triangulation is aligned wit te partition of Ω, tat means tat te boundary Ω i is made up of faces from simplices in Let g D, a inite Element approximation of g D on Γ D en te solution of te discrete problem u satisfies u g D, + V and (323) Ω k(x) u (x) v (x) dx = Ω f(x) v (x) dx + g N (x) v (x) dx v V Γ N We disregard problems arising from te approximation of non-omogeneous boundary conditions, tat means we set g D = g D, on Γ D and assume tat g N is piecewise constant on faces of simplices wit meas d 1 ( Γ N ) > 0 As Γ D as positive measure uniqueness and solvability of problems (322) and (323) is given by Riesz s teorem

3 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD Assumptions and Notation e space dimension will be denoted by d We use te terminus 2D case or 3D case to indicate tat d =2or d =3 Unless stated oterwise all results cover te 2D and te 3D case simultaneously We define te weigted (semi-)norm v 2 kh 1 (Ω ) := k(x)( v(x)) 2 dx Ω In te following all integrals are over te space variable x and for te sake of sortness we do not explicitly denote te dependence on x e triangulation consists of simplices wic intersect at most at a common face, edge, or vertex Suc a triangulation is called admissible [15] We make te following assumptions for te discrete problem (323): e family of triangulations { } is sape regular but not necessarily uniform [15] e diameter of a simplex is denoted by o ease notation we may drop te subindex and take te value of from te simplex under consideration We will use also te term mes or grid instead of triangulation e triangulation defines a set Ω, wic coincides wit Ω urter Γ D and Γ N consists of wole faces of simplices from o avoid ambiguity we suppose tat Γ D is a closed set Definition 31 or functions a, b depending on te data k, f, u, u,g N,g D wit values in R 1 we use te notation (324) a b to indicate, tat it olds a cbfor a furter not specified constant c, tat does not depend on te data but only on te sape regularity parameter of te finite element mes e bound (324) will be called robust We write a b, if it olds a b and b a e set of nodes of te triangulation will be denoted by N We denote by all faces from simplices from tat are not contained in Γ D e set of all edges will be denoted by E All simplices, faces, and edges E E are closed sets or a subset S Ω we denote by N (S) te nodes contained in S and wit E (S) te edges in S or a node x resp an edge E or face we define ω x resp ω E or ω as te union of all simplices, wic ave x resp E or in common e value of k on a simplex will be denoted by k or a face denote by a simplex from ω suc tat k =max ω k We define k := ω k Clearly k k We define a simplex-wise constant approximation f of f or instance one may coose te average on te simplex, f := 1 f

4 42 33 INIE ELEMEN MEHOD ON UNIORM GRIDS Definition 32 Let be not contained in Γ N Denote by n and n te outward normal of resp Let u be te solution of problem (323) e jump of te normal fluxes across te face is defined as u u j := k + k n n Let be a face on te Neumann boundary Denote by n te outward normal of Ω e jump of te normal fluxes across tis face is defined as j := g N k u n e inite Element sape function taking on te value 1 at te node x i and vanising on all oter nodes of te triangulation are denoted by λ i We define so-called bubble functions ese are non-negative sape functions wit small support and wic are not contained in V Anelement bubble function can be defined as φ := (d+1) d+1 x i λ i, were te product is taken over all nodes of x i on Aface bubble function φ for te face will ave ω as support We define φ = d d x i λ i, were te product is taken over all nodes x i on te face 33 inite Element Metod on uniform grids Let us assume in tis section tat te mes is quasi-uniform at means, tat te local cell diameter ( ) is of te same order for eac simplex en te approximation error can be measured in terms of te mes diameter Usually te convergence rate of te solution u of (323) to te solution u of (322) depends on te number s wic yields global regularity u H 1+s (Ω) We sow tat in te 2D case it is te piecewise regularity u H 1+s (Ω i ) and not te global regularity tat bounds te convergence rate or a proof in 3D see [11] Recall tat regularity results from [33] [31] sow u H 1+s ( Ω) for a certain s>0 e dependence of s from δ is given in 2D in eorem 217 By Sobolev embeddings we know tat u is continuous in Ω [1] us, te interpolation operator I N : V V wic is given by taking te values in te nodal points is well defined Exploiting arguments given in Example 3 in [25], one as te following interpolation results in fractional Sobolev spaces H 1+s,s>0: (331) u I N (u) H 1 ( ) s u H 1+s ( ) Combining tis bound wit Galerkin ortogonality sows an estimate in terms of piecewise contributions for u H 1+s (Ω i ): (332) u u 2 kh 1 (Ω) u I N(u) 2 kh 1 (Ω) = n n k i u I N (u) 2 H 1 (Ω i ) 2s k i u 2 H 1+s (Ω i ) l=1 l=1

5 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 43 is bound sows tat inite Elements make in a certain sense use of piecewise regularity As global regularity is restricted to H 3/2 ε (Ω), for any ε>0, (see section 232), it follows, tat for s>1/2 te convergence is bounded by piecewise regularity and not by global regularity As te regularity parameter s, wic depends on k, can be arbitrary small, te convergence rate can deteriorate Remark 31 ake te singular function ψ 2 from example 22 on page 16 and coose λ (0, 1] Using approximation results from [47, p265] [60] we see tat on a simplex, containing te singular point, te convergence of te solution u of problem (323) to te solution u of problem (322) could be arbitrarily bad c(u) λ u u kh 1 ( ) λ ε C(u) ε >0 is is also true for polynomial inite Elements of iger order [20] In view of iger order convergence in regions were te solution is in H 2, te singularity will asymptotically dominate on uniform meses leading to a global convergence rat of order O( λ ε ) for any ε>0 In order to enance convergence for solutions wit poor regularity, several tecniques ave been developed One possibility is te enricment of te Galerkin space wit functions, tat approximate te singular functions [56] See also [20] for a modified procedure Anoter approac consists in refining te mes around te singularities 34 inite Element Metod on adapted grids Let us start wit an initial mes By refinement of te initial mes around a singular point we understand te construction of a triangulation wit additional degrees of freedom around te singular point Here we differ between an approac, were te degree of te singularity sould be known a-priorily, and a competing approac, were no a-priori knowledge is required and te mes is refined on te basis of a-posteriori error estimators e a-priori approac makes use of te fact tat te form of te singular solution r λ s λ (ϕ) is known Here (r, ϕ) are polar coordinates wit respect to a singular point A sape regular triangulation is constructed in suc a way tat te local mes diameter beaves like Hr 1 µ,µ < λ, [47] [6] Here H is a parameter and can be seen as global mes size Additionally, te simplex containing te singular point sould ave a diameter of order H 1/µ In order to construct suc meses, λ sould be known or approximated [32] Witin tis setting te error reduction proceeds wit optimal order O(H) In 3D te parameter r is te distance to te singular edge and anisotropic simplices tat are stretced in direction of te edge may be used See [5] for te case of a constant diffusion coefficient k(x) 1 e drawback of a-priori refinement is not only tat λ sould be known but also tat tere is no control of te refinement dept (te parameter H) in te vicinity of a singular point: It may be te case tat in a particular situation te contribution of a singular function is zero or very small, wen compared to te regular part or to oter singularities en tere is no need of refining te singularity

6 44 35 INERPOLAION OPERAORS Besides varying te local grid-size one can increase te degree of te polynomial sape functions is leads to te p-version, developed for te case k =1[7] [8] [53] e p-version as been extended to non-polynomial sape functions for interface problems in [48] Here again te metod suffers from te drawbacks of a-priori refinement and te convergence rate depends on te approximation of λ But te metod leads to a very ig rate of convergence even in case of very low regularity An alternative a-posteriori approac is te construction of a new refined mes by subdividing simplices on wic te error is large into smaller ones is corresponds to an introduction of new degrees of freedom Since te error itself is not known, one uses a-posteriori error estimators η, wic sould reflect te beaviour of te error: η u u kh 1 ( ) e a-posteriori error estimators can be calculated wit low numerical effort on te basis of known data Suc estimators ave been derived for te Laplace equation An overview is given in [57] or te interface problem tere are recent independent articles [21] [11] and [49] ese articles impose different restrictions on te structure of k In [21] te case of essentially two subdomains is treated In [11] a special criterion concerning te distribution of diffusion coefficients is used; see remarks 33 and 34 In te present capter weaker assumptions on k are posed e reason for te restriction of te diffusion coefficient is tat we will make use of interpolation operators wic are robust in respective function spaces In te next section we argue tat te restriction imposed on k in te present work are necessary for te existence of suc robust interpolation operators 35 Interpolation operators 351 A non-robust interpolation operator In tis section we use te sorter terminus coefficient for te diffusion coefficient k or simplices and for faces we need estimates of te form (351) u I L (u) L 2 ( ) u kh 1 ( ω ) u I L (u) L 2 ( ) 1/2 u kh 1 ( ω ), were I L : V V is an interpolation operator and ω, ω consists of some neigbouring simplices of and It is important tat no additional factors depending on k do not enter into te bounds (351) An interpolation operator fulfilling te bounds (351) will be called robust e Clement interpolation operator I C [16] [57] fulfills only te non-robust bounds Lemma 31 Let δ k δ 1 or any and any te non-robust bounds (352) u I C (u) L 2 ( ) δ 1 u kh 1 (ˆω ), u I C (u) L 2 ( ) 1/2 δ 1 u kh 1 (ˆω ),

7 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 45 old Here ˆω, ˆω denote te union of simplices from tat ave a node wit resp in common ese bounds are optimal wit respect to δ PROO e proof is an easy consequence of te properties of te Clement interpolation operator u I C (u) L 2 ( ) u H 1 (ˆω ) u I C (u) L 2 ( ) 1/2 u H 1 (ˆω ) Denote by k min,k max te lower and upper bound of k on ˆω It is easy to see tat u 2 H 1 ( ω ) u 2 H k 1 ( ) = 1 u 2 kh min k 1 (ˆω ) k 1 k max u 2 kh min k 1 (ˆω ) min ˆω k e relation k max /k min δ 2 finises te proof of te first inequality e second inequality is sown in te same way 352 e quasi-monotonicity condition revisited Robust interpolation operators ave been derived under certain restrictions on te coefficient k [24] [13] or some special configurations of diffusion coefficients it as been sown in [62] tat tere are no robust interpolation operators In 2D regard a ceckerboard like distribution of coefficients from {1,ε} In 3D two cubes touc at a vertex or on an edge Inside te cubes te diffusion is k 1 =1and in te remaining part of te domain k 2 = ε ollowing [24] we introduce te class of quasi-monotone coefficients Witin tis class one can define robust interpolation operators [24] or 2D te class of quasi-monotone coefficients was already defined in Definition 24 in capter 2 and tis class coincides wit te one defined below Definition 33 or a node x N ( Ω) we denote by x a simplex from ω x were te coefficient k acieves te maximum for ω x Definition 34 Coose a node x N ( Ω) e distribution of coefficients k, ω x will be called quasi-monotone wit respect to a node x of a triangulation and te part of te boundary Γ D Ω if te following conditions are fulfilled or eac simplex ω x tere exists a Lipscitz set ω,x,qm containing only simplices from ω x suc tat if x N ( Ω/ Γ D ) ten x ω,x,qm and k k ω,x,qm if x N ( Γ D ) ten ω,x,qm, meas d 1 ( ω,x,qm Γ D ) > 0 and k k ω,x,qm

8 46 35 INERPOLAION OPERAORS x 1 x igure 31: e distribution of coefficients k, ω x is quasi-monotone wit respect to te node x on te left figure but not on te rigt one e simplex is colored dark and te set ω,x,qm is colored wit different levels of grey in te left picture Γ D x Γ D x igure 32: e distribution of coefficients k, ω x is quasi-monotone wit respect to te node x N (Γ D ) on te left figure but not on te rigt e simplex is colored dark and te set ω,x,qm is colored wit different levels of grey in te left figure One cecks tat te definition does not depend on te coice of x urter it is important to note tat tis definition is in 2D also independent of te triangulation and describes a property of te diffusion coefficients If tere is no danger of confusion, we will simply say tat te distribution of coefficients is quasi-monotone wit respect to a node x if te above definition is fulfilled for x We say tat te distribution of coefficients k, is quasi-monotone if it is quasimonotone wit respect to all nodes of N Here we use te definition of Lipscitz domains as given in [15] It follows tat te union of two simplices from saring a node is a Lipscitz domain if and only if tey sare a face We illustrate te quasi-monotonicity condition for an interior node (igure 31) and for a node x Γ D (igure 32) Note tat a cecker board like distribution of te coefficients is not quasi-monotone Remark 32 Definition 34 can be formulated in a more intuitive way We give only te idea or a node x N ( Ω/ Γ D ) we demand tat te trace of k on a small spere B around x as

9 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 47 only one local maximum We say tat a local maximum is attained on B Ω i,ifk i >k j for all adjacent subdomains Ω j : meas d 1 ( Ω i Ω j ) > 0 wit node x Ω j Ifx N ( Γ D ) eac local maximum is adjacent to Γ D Remark 33 Under te following restrictions any distribution of coefficients is quasi-monotone Let x Ω a point, denote by n te number of subdomain Ω i to wose closure x belongs and by m te number of boundary types from Γ D, Γ N to wic x belongs In oter words set m =1if pure Diriclet or pure Neumann boundary conditions are imposed on x Ω, set m =0if x is an interior point and oterwise m =2Ifin2Dn is restricted by n 3 m, ten te distribution of coefficients k, ω x, is quasi-monotone for any values of k A similar bound olds in te 3D-case If n 2 m te distribution of te coefficients is quasi-monotone independent of te values of k One cecks tat tese restrictions on n are sarp, except for te case m =2 o see tis in case of an interior node x in 3D regard to cubes Ω 1, Ω 2, wic touc only in x e diffusion takes on te value 100 in te cubes and te value 1 in te remaining part Ω 3 en n =3,m =0 and te distribution of te diffusion coefficients is not quasi-monotone a k 2 =10 k 1 =1 k 3 = 100 k 4 =10 e igure 33: quasi-monotone diffusion coefficients wic do not fulfill condition from [11] e following restriction of te diffusion coefficient as been used in [11] to derive robust interpolation operators Remark 34 Suppose tat te coefficients k, ω x, are distributed in suc a way tat for any two simplices b, e ω x tere are simplices i ω x,i=1,, n, were 1 = a, n = b and meas d 1 ( i i+1 ) > 0 for i = 1,, n 1 Suppose furter tat te sequence k 1,k 2,, k n 1,k n is monotone en it is not ard to see tat te distribution of coefficients k, ω x is quasi-monotone wit respect to a node x if te space dimension is 2 or if x does not belong to Γ D Ifin3Da node belongs to Γ D tis condition is not sufficient to define robust interpolation operators o

10 48 35 INERPOLAION OPERAORS see tis, regard te 3D example of a cube Ω 1 toucing te Diriclet boundary Γ D = Ω at one vertex only Define Ω 2 to be te remaining part of te domain and set k 1 := 100,k 2 := 1 as proposed in [62] We sow tat te above condition is stronger tan quasi-monotonicity in te 2D case or if x N ( Ω/ Γ D ) Define in 2D four coefficients wic are numbered clockwise and wic take on te value k 1 =1,k 2 =10,k 3 = 100,k 4 =10(igure 352) aking b = 4, b = e te above condition is not fulfilled but te distribution of coefficients k i is quasi-monotone In 3D proceed similarly Remark 35 Using te counter examples from Xu [62] one notices tat quasi-monotonically distributed coefficients are te largest class of coefficients for wic robust interpolation operators exists is is easily seen from remark 32 e counter examples in [62] exploit te fact tat tere is more tan one local maximum of te diffusion coefficient wen restricted to a small spere around a vertex in te sense of remark 32 We say tat a triangulation is a refined triangulation of if for te according inite Element spaces olds V V In 2D quasi-monotonicity is preserved during refinement of a triangulation E 100 igure 34: tree orizontal layers, in te middle layer tere is a ceckerboard-like pattern of diffusion coefficients 1, 10 around te edge E; for a new node on E quasimonotonicity will be violated In 3D tis is not true or example regard te case of a domain Ω=( 1, 1) ( 1, 1) ( 1, 2) subdived into tree orizontal layers of equal size as sown in igure 352 Eac layer is again subdivided into four similar cubes In te bottom and upper layer te coefficient is 100 or te four cubes saring te edge E given by te points (0, 0, 0) and (0, 0, 1) te coefficients 1 and 10 are distributed alternately Given Diriclet boundary conditions on Ω and given a triangulation wit nodes on vertices of te 12 cubes, te distribution of coefficients is quasi-monotone But if in te course of refinement a new node is created on te edge E, te quasi-monotonicity condition is violated for tis node o assure quasi-monotonicity also for triangulations obtained by refinement we introduce te following

11 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 49 Definition 35 Let d =3 Coose an edge E E e distribution of te coefficients k, ω E will be called quasi-monotone wit respect to te edge E of a triangulation and te part of te boundary Γ D Ω if following conditions are fulfilled: Denote by E a simplex from ω E were k acieves te maximum in ω E or eac simplex ω E tere exist a Lipscitz set ω,e,qm containing only simplices from ω E, suc tat if E E ( Ω/ Γ D ) ten E ω,e,qm and k k ω,e,qm if E E ( Ω/ Γ D ) ten ω,e,qm, meas 2 ( ω,e,qm Γ D ) > 0 and k k ω,e,qm We say tat te distribution of coefficients k, is quasi-monotone wit respect to edges of if te above condition olds for all edges E E o illustrate tis condition in 3D, denote by G E a 1-dimensional spere perpendicular to E, wit center on E and contained in ω E is definition states tat for interior edges E te coefficient function as only one local maximum on G E Remark 36 If in 3D te distribution of coefficients k,, is quasi-monotone and additionally quasi-monotone wit respect to edges of, ten te distribution of coefficients k,, is quasi-monotone for any refined triangulation 353 A robust interpolation operator In te case of a mass term occuring in te elliptic equation one may need an interpolation operator wic is continuous in L 2 (Ω) Suc stability does not old in te case of te interpolation operator defined in [24] A furter difference is tat we allow also for mixed boundary conditions Our interpolation operator differs from te one presented in [24] only for nodes on te boundary Ω e same operator was proposed independently in [11] Let te distribution of coefficients k, be quasi-monotone or a simplex define a set containing some neigbouring simplices of ω := ω,x,qm x N ( ) or a face define ω by substituting in te above definition wit e quasi-monotonicity condition implies ten k k ω Remember tat x is defined in definition (33) e interpolation operator is defined by I L u := λ i p xi, were p xi := 1 u, x i N (Ω Γ N ) xi xi x i N (Ω Γ N )

12 50 35 INERPOLAION OPERAORS and λ i are te finite element sape functions of V Hence I L : V V or convenience define p x := 0 for nodal points x Γ D so tat in fact I L u := x i λ i p xi were te sum is taken over N ( Ω) Alternatively I L can be defined by substituting in te definition of p xi te term xi by te union of simplices were k takes te maximum in ω x We need a scaled version of a standard trace inequality It states tat Lemma 32 Let Ω 0 R d,d =2, 3, be a domain wit diameter and Lipscitz boundary Let be a subset of Ω 0 wit positive measure en for v H 1 (Ω 0 ) v 2 L 2 ( ) 1 v 2 L 2 (Ω 0 ) + v 2 H 1 (Ω 0 ) PROO is is a refined version of a standard trace inequality [29] for domains wit diameter O(1) e constant in te bound depends on te Lipscitz constant of Ω 0 Here we state te main result in tis section Lemma 33 Let d =2, 3 Let u V Coose a simplex and a face Let te distribution of coefficients k, ω x be quasi-monotone wit respect to all nodes x of and en te following bounds old for any v V (353) (354) (355) (356) I L (v) L 2 ( ) v L 2 ( ω ) v I L (v) L 2 ( ) v H 1 ( ω ) k 1/2 v kh 1 ( ω ) v I L (v) H 1 ( ) v H 1 ( ω ) k 1/2 v H 1 ( ω ) v I L (v) L 2 ( ) 1/2 v kh 1 ( ω ) PROO e proof is similar to tat in [24] As te definition of I L contains only integrals on simplices but no integrals on faces it is possible to bound te L 2 -norm of I L in terms of te L 2 -norm Coose a simplex and number its nodes wit x i,i=0,, d Let x N (Ω Γ N ) Note tat p x can be written as P x (v) were P x is te L 2 -ortogonal projection on constant functions in L 2 ( x ) Exploiting te property of tis projection it yields for nodes x i N (/Γ D ) and any c R (357) p xi c 2 L 2 ( ) p xi c 2 L 2 ( xi ) = P xi (v c) 2 L 2 ( xi ) urter, we use te decompositions d (358) v = λ i v and I L (v) = i=0 We conclude from (357) wit c =0and (358) tat v c 2 L 2 ( xi ) d λ i p xi i=0 I L (v) 2 L 2 ( ) d i=0 p xi 2 L 2 ( ) d i=0 v 2 L 2 ( xi ) v 2 L 2 ( ω )

13 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 51 is sows (353) Now let us prove (354) rom (358) we obtain (359) v I L (v) 2 L 2 ( ) d i=0 λ i (v p xi ) 2 L 2 ( ) d i=0 v p xi 2 L 2 ( ) Inequality (357) applied to nodes x i N (/Γ D ) yields for any c R v p xi 2 L 2 ( ) v c 2 L 2 ( ) + p xi c 2 L 2 ( ) v c 2 L 2 ( ) + v c 2 L 2 ( xi ) Recall tat from definition x ω,x,qm ω We use te last inequality and apply te Poincaré inequality [4] to te Lipscitz set ω,xi,qm and arbitrary c to obtain (3510) v p xi 2 L 2 ( ) v c 2 L 2 ( ω,xi,qm) 2 v 2 H 1 ( ω,xi,qm) or nodes x i from lying on te Diriclet boundary we use p xi =0and te fact tat v vanises on ω,x,qm Γ D e Poincaré-riedrics inequality [4] yields (3511) v p xi 2 L 2 ( ) v 2 L 2 ( ω,xi,qm) 2 v 2 H 1 ( ω,xi,qm) Collecting inequalities (3510) and (3511), togeter wit (359), sows (3512) v I L (v) 2 L 2 ( ) 2 v 2 H 1 ( ω ) It remains to use te quasi-monotonicity condition wic states (3513) k k ω Wit tis bound we prove (3514) v 2 H 1 ( ω ) = k 1 v 2 kh 1 ( ) k 1 v 2 kh 1 ( ω ) ω wic yields due to (3512) assertion (354) or sowing (355) we use as before (358) to conclude v I L (v) 2 H 1 ( ) d i=0 λ i (v p xi ) 2 H 1 ( ) e properties of te sape functions λ i imply ten (3515) λ i (v p xi ) 2 H 1 ( ) ( λ i )(v p xi ) 2 L 2 ( ) + λ i (v p xi ) 2 L 2 ( ) 2 v p xi 2 L 2 ( ) + v 2 H 1 ( ) We combine again (3510) and (3511) to bound d (3516) 2 v p xi 2 L 2 ( ) v 2 H 1 ( ω ) i=0

14 52 35 INERPOLAION OPERAORS rom te last tree inequalities and (3513) follows now (355) e trace inequality from Lemma 32 and inequalities (354), (355) sow v I L (v) 2 L 2 ( ) 1 v I L (v) 2 L 2 ( ) + v I L (v) 2 H 1 ( ) v 2 H 1 ( ) Multiplication wit k k proves due to (3513) wit ω substituted by ω te last assertion (356) 354 Does regularity imply interpolation properties? In tis section we set d =2 We saw tat te property of quasi-monotonicity implies on te one and regularity H 5/4 (section 254) and on te oter and approximation properties of inite Elements in norms depending on te coefficient k (see inequality (355)) It is interesting weter tere is a connection between regularity H 5/4 and tese approximation properties is open question will be addressed in te following In classical interpolation results, as for instance for te Lagrange interpolation operator, regularity H d/2+ε (for some ε>0) is needed [15] and te stronger semi -norm H d/2+ε will appear on te rigt and side of te interpolation inequality is is a difference to our approac since we want to use only te weaker norm L 2 (Ω) ix an interior singular point x 0 and denote by B a ball centered at x 0 and contained in ω x and wit diameter approximately Suppose tat B contains no oter singular point Our aim is to ave an interpolation result of te analogon of inequality 354 in following form (3517) (v v ) L 2 (B) cλ 1/2 v L 2 (B) for functions v V of te special form v := u u were u, u, are solutions of problems (322),(323) and u H 1+s (Ω i ) for any 0 <s<λ Here v V is an approximation of v and c does not depend on te data k nor on v, u or u Comparison wit inequality 354 sows an additional factor λ 1/2 tat depends on te regularity of v and tus on k is factor is necessary since it as been sown in [Xu] tat inequalities like (3517) do not old for arbitrary v V and v V witout additional factors depending on v e bound (3517) is stronger tan (352) since we conclude from eorem 216 tat δ λ and ence λ 1/2 δ 1/2 <δ 1 Setting for a moment u =0, v =0and v = ψ 2, were ψ 2 is defined in example 22, we see tat te factor λ 1/2 is necessary e same is true for functions v wit a small norm wit respect to v We ave to pay for decreasing regularity wit a iger factor We do not know if an estimate like (3517) olds but we want to give some ideas ow to ceck it, if it olds Defining v (x 0 )=v(x 0 ) and v (x i )=p xi for points different from x 0 and using tecniques from te proof of Lemma 33 inequality (3517) may be sown, if we were able to sow (3518) (u u ) L 2 (B) λ 1/2 (u u ) L 2 (B) Suppose tat f B =0 is implies tat u is piecewise armonic on B Ω i We now use tecniques based on decomposition into ortogonal bases, see for instance [56]

15 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 53 In tis subsection we use te notation λ i,i=1, 2,for te eigenvalues λ 2 i of te Sturm- Liouville eigenvalue problem (241), (242) is sall not lead to confusion wit te similar notation for te inite Element sape functions Denote by s λi,i =1, 2, te according eigenfunctions erefore, functions s λi,s λj,i j are ortogonal in te scalar products induced by te norm ( ) L 2 ( B) and te norm ( ) L 2 ( B) Using te expansion u B = i a is λi,a i R and te fact tat r λ i s λi is piecewise armonic and satisfies te interface conditions we can expand u in te sequence [56] (3519) u = i a i r λ i s λi Regularity of u is restricted by regularity of functions r λ i s λi and we observe tat it olds piecewise H 1+s -regularity for any s<λwere λ =min i {λ i } Ortogonality of te functions s λi implies ortogonality of functions te r λ i s λi in te scalar products defined by te norms ( ) L 2 (B) and ( ) L 2 (B) urter we use (r λ i s λi ) L 2 (B) λ 1/2 i ( r λ i s λi ) L 2 (B) We may prove now (3520) u L 2 (B) λ 1/2 u L 2 (B) by expanding u in te ortogonal basis given in (3519), applying Parseval s identity and sowing appropriate inequalities for eac of te functions r λ i s λi Using standard inite Element tecniques it is not ard to prove te estimate (3521) u L 2 (B) λ 1/2 u L 2 (B) If one wants to combine (3520), (3521) to sow (3518) one would need a sarpened Caucy-Scwarz inequality between te Spaces V and V equipped wit te norm L 2 (B) (or an additional condition for te norm L 2 (B) ) We do not know ow to derive suc an inequality or weter it olds at all Anoter attempt in proving te relation (3517) could be to proceed as in te proof of te Poincaré inequality wic relies on compact embeddings L 2 H 1 erefore, it suffices to ceck weter te function v =1defined on (0, 1) is not contained in te closure of functions r λ i, 0 <λ 0 λ i,i=1, 2,defined on te interval (0, 1) in te H 1/2 - semi-norm and does belong to te closure of tat functions in L 2 (0, 1) Note tat r λ i are te traces of functions r λ i s λi (ϕ) But tis attempt fails as one can sow tat te function v belongs to te closure in H 1/2 (0, 1) iff it belongs to te closure in L 2 (0, 1) e proof of tis fact is not straigtforward It uses ideas of te proof of Müntz teorem [19, p 174] Instead of using in te proof te Hilbert space H 1/2 (0, 1) we can use te space H 1 (B 0 ) were B 0 is te unit ball e definition of te functions r λ i remains te same 36 Residual based error estimators 361 eoretical basis for a-posteriori error estimators We will use te interpolation operator defined in section 35 If te diffusion coefficients k,, are distributed quasi-monotonically, we can exploit te approxima-

16 54 36 RESIDUAL BASED ERROR ESIMAORS tion estimates (351) See remarks 33, 34 in section 35 for sufficient conditions for te quasi-monotonicity If te distribution of diffusion coefficients is not quasi-monotone wit respect to some nodes, ten tere are no robust interpolation operators satisfying equations (351) In tis case we ave to admit a constant δ from relation (321) We define te residual r(u) V for te solution u of (322) and u of equation (323): r(u)(v) := k (u u ) v for v V Ω e following decomposition of te residual will be used in te derivation of an upper bound for te error Lemma 34 or any v V and any v V we ave te following representation of te residual: (361) k (u u ) v = f (v v )+ j (v v ) Ω PROO Recall tat we suppose g D = g D,, tat means tat problem (322) and (323) fulfill te same Diriclet conditions Integration by parts allows for splitting te residual into local contributions We use Galerkin ortogonality togeter wit te fact tat u vanises for linear functions Note tat in te definition of j we ave included te Neumann boundary data 362 A residual based error estimator Extending an estimator from [57] we define a residual based estimator η R e global estimator η R consists of te sum of local estimators η R, Definition 36 ηr 2 := ηr, 2 η 2 R, := 2 k f 2 L 2 ( ) + /Γ D k j 2 L 2 ( ) e next teorem is te main result in tis section We sow te estimator to be reliable and efficient A robust upper bounds olds in te case of quasi-monotone diffusion coefficients Oterwise additional constants enter in te upper bound for te estimator, see remark 37 eorem 35 Let d =2, 3 and g D = g D, and let g N be piecewise constant on faces (Γ N ) If te distribution of te diffusion coefficients k,, is quasi-monotone, ten for te solution u of equation (322) and u of equation (323) it olds tat te estimator η R is globally reliable, tat is (362) u u 2 kh 1 (Ω) ηr, f f 2 L k 2 ( )

17 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 55 Witout any assumptions about te distribution of te diffusion coefficients te estimator η R is locally efficient, tat is for any simplex ηr, 2 u u 2 kh 1 (ω ) + 2 f f 2 L 2 ( ), ω were ω contains all simplices saring a face wit e constants in tese bounds neiter depend on te diffusion k nor on oter data, but only on te sape regularity parameter of In te 2D-case it suffices to demand in eorem 35 quasi-monotonicity for te initial grid But remember tat te quasi-monotonicity is preserved during refinement in 3D only under an additional condition given in Definition 35 Corollary 31 Let d =3and let an initial triangulation 0 be given e distribution of diffusion coefficients k, 0, is quasi-monotone and additionally te distribution of diffusion coefficients k, 0, is quasi-monotone wit respect to edges of E 0 en for eac triangulation obtained by refining 0 (tat means for te corresponding inite Element spaces olds V V 0 ) te error estimator η R is reliable u u 2 kh 1 (Ω) η 2 R, + k 2 k f f 2 L 2 ( ) Remark 37 Wit te bounds δ k(x) δ 1 te non robust upper bound (363) u u 2 kh 1 (Ω) δ 2 ηr, f f 2 L k 2 ( ) olds or a proof proceed as in [57] for te case k =1 Remark 38 In [11] similar error estimates were derived independently for te class of diffusion coefficients, defined in remark 34, wic is smaller tan te class of quasi-monotone diffusion coefficients See also te independent article [21], were te case of essentially two subdomains was considered PROO of eorem 35 e tecniques used in te proof are essentially tose of [57] Define v = u u, were u and u are solutions of (322) and (323) and observe tat v vanises on Γ D Let v := I L (v) Reliability will be sown using te representation of te residual in Lemma 34 and Lemma 33 from section 35 for bounding te terms v v By Lemma 34 we split te residual into contributions f (v v ) and j (v v ) We use Lemma 33 from section 35 to bound (364) f(v v ) f L 2 ( ) v v L 2 ( ) f L 2 ( ) v kh 1 ( ω ) In a second step we estimate te contributions from te terms j Using te approximation inequality (356) from Lemma 33 from section 35 we ave te local estimate j (v v ) j L 2 ( ) v v L 2 ( ) (365) ( ) 1/2 j k L 2 ( ) v kh 1 ( ω )

18 56 36 RESIDUAL BASED ERROR ESIMAORS Now we use (361) togeter wit te bounds (364), (365) and apply te Caucy- Scwarz inequality urter we make use of te fact tat eac simplex is covered by finite number of ω or ω and obtain Ω k (u u ) v = v 2 kh 1 (Ω) v kh 1 (Ω) 2 f 2 L k 2 ( ) + /Γ D k j 2 L 2 ( ) 1/2 Cancelation and te triangle inequality f L 2 ( ) f L 2 ( ) + f f L 2 ( ) finis te proof of (362) e proof of te lower bound goes in two steps irst we estimate te element residual We denote by φ an element bubble function vanising outside as defined in section 322 Note tat since f is a constant on eac simplex, te local equivalences f φ H 1 ( ) f L 2 ( ) 1, f φ L 2 ( ) f L 2 ( ) follow directly from te relation φ H 1 ( ) 1 1 L 2 ( ) and φ L 2 ( ) 1 L 2 ( ) Using v = f φ as a test function and seting v = 0 in (361) te Caucy-Scwarz inequality and te local equivalences imply f 2 L 2 ( ) f (f φ ) = k (u u ) (f φ ) (f f )(f φ ) ( ) 1 u u kh1( ) + f f L2( ) f L 2 ( ) Simplifying te last expression we get an upper bound (366) f L 2 ( ) u u kh 1 ( ) + f f L 2 ( ) o complete te estimate from below we need to estimate te jump term We will exploit local equivalences of te type j φ L 2 ( ) 1/2 j L 2 ( ), j φ H 1 ( ) 1/2 j L 2 ( ) ese equivalences follow from φ L 2 ( ) 1/2 1 L 2 ( ) and φ H 1 ( ) 1/2 1 L 2 ( )

19 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD 57 We insert v = j φ as a test function and let v =0in (361) and obtain togeter wit te above local equivalences j 2 L 2 ( ) j (j φ ) = f (j φ ) (f f )(j φ )+ k (u u ) (j φ ) ω ω ω { ( ) f L 2 ( ) + f f L 2 ( ) j L 2 ( ) 1/2 ω +k u u H 1 ( ) j L 2 ( ) 1/2} Cancelation and insertion of (366) into te last rigt and side gives (367) 1/2 j L 2 ( ) { } f f L 2 ( ) + u u kh 1 ( ) ω f f L 2 (ω ) + u u kh 1 (ω ) Combing (366) and (367) we obtain te upper bound for η R, η 2 R, u u 2 kh 1 (ω ) + ω 2 k f f 2 L 2 ( ) 363 Error estimators based on local problems In te literature one finds estimators based on te solution of local problems [57] Let te domain Ω be covered by patces of simplices On eac patc one defines a Galerkin space of bubble functions and solves witin tis space a local analogon of te continuous problem e energy norm of te resulting solution is taken as an error estimator e local problems can be classified into Diriclet and Neumann problems Error estimators based on local Diriclet problems Here we present an approac wit Diriclet boundary conditions on a patc consisting of two neigbouring simplices or a face let and be simplices saring te face If (Γ N ) regard only one simplex wit face e Galerkin space consists of two element bubble functions φ,φ vanising outside resp and one bubble function aligned wit te face φ vanising outside ω Let V D be te space spanned by tree bubble functions φ,φ,φ Suc bubble functions are defined in subsection 322 We seek v D V D fulfilling: (368) k v D φ = f φ + g N φ k u φ φ V D ω ω Γ N ω

20 58 36 RESIDUAL BASED ERROR ESIMAORS Definition 37 or a face we define η D, := v D kh 1 (ω ) Similar estimators ave been proposed in te context of ierarcical bases in [12], see also section 371 We sow tat te residual based estimator η R and te estimator η D are equivalent eorem 36 Let d =2, 3 or eac face and neigbouring simplices, we ave η D, η R, + η R, and η R, /Γ D η D, PROO e proof uses tecniques from [57] developed for te case k =1adapted to te case of piecewise constant diffusion coefficients as done in te proof of eorem 35 Error estimators based on a local Neumann problems It may be desirable to construct estimators based on local problems wit just one simplex as support for te bubble functions en one as to impose Neumann boundary conditions or varying coefficients it is not straigtforward wat Neumann boundary conditions to impose in order to keep te resulting estimator robust We propose an estimator wit one bubble function φ per face of and one φ for te simplex as done in te estimator η N in [57] e Galerkin space spanned by tese functions is denoted by V N We seek v N V N satisfying: (369) k v N φ = f φ /Γ D ( k k ) 1/2 j φ φ V N e estimator η N will ten be defined as: Definition 38 η N, := v N kh 1 ( ) Again we can sow an equivalence eorem 37 e estimators η R, and η N, are locally equivalent: ηn, 2 η2 R, PROO e proof is done as te proof of eorem 36

21 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD Oter estimators 371 Estimators based on ierarcical bases In distinction from residual based error estimators, were one needs interpolation results for te derivation of upper bounds for te error, tere is an alternative approac based on ierarcical bases, see [12] [9] Here te upper bound is sown by te so-called saturation assumption e analysis in [12] was done for te case k 1 but carries over directly to te case of discontinuous diffusion coefficient Let V Q V were Q is for example te space of piecewise quadratic inite Elements Q = V V Here V is called te ierarcical extension We define u Q as te solution of te variational problem wit Galerkin space Q: u Q g D, Q and (371) k u Q v Q = fv Q + g N v Q v Q Q Ω Ω Ω We define te saturation assumption Definition 39 We say tat te saturation assumption is fulfilled if tere exists a number β (0, 1) suc tat for te solutions u, u,u Q of problems (322) (323) (371) olds u u Q kh 1 (Ω) β u u kh 1 (Ω) and β does not depend on te data f,g D,,g N or k e ierarcical extension V is spanned by face bubble functions φ for faces We define te local error estimator for a face ) 2 ηh, (ω 2 = fφ j φ φ 2 kh 1 (ω ) and te global error estimator η 2 H := η 2 H, ollowing [12] one can sow Lemma 38 or solutions u, u of problems (322) (323) olds te lower bound ηh 2 u u 2 kh 1 (Ω) e saturation assumption is equivalent to te upper bound u u 2 kh 1 (Ω) ηh 2

22 60 37 OHER ESIMAORS Similar estimators can be constructed for different spaces Q V e problem is to prove te saturation assumption or te case of a constant diffusion coefficient k tis as been done recently using a-posteriori error analysis tecniques [23] ere it is sown tat a small ratio between te oscillation f f 2 L 2 ( ) and f 2 L 2 (Ω) implies te saturation assumption However, for te case of a varying k until now tere is no result sowing te saturation assumption on te basis of a-priori data f If one is solely interested in te saturation assumption, one can use reliability of te adjoint error estimator to prove te saturation assumption [12] is can be done if te space V contains for eac simplex a sape function wit support contained in as ϕ [12] See [21] for connections between so called a-posteriori saturation assumption tat drives te quality of te approximation of te data and te refinement strategy and between te above defined saturation assumption 372 A Zienkiewicz-Zu like estimator Besides residual based and ierarcical based error estimators tere is anoter approac based on averaging tecniques originating from Zienkiewicz and Zu [63] [57] is approac is popular among engineers and is validated for te case of a constant diffusion coefficient In tis section we want to provide an argumentation tat it may be not possible to adapt tis kind of estimators to te general case of discontinuous diffusion coefficients in a robust way We do tis in te ligt tat tere are applications using averaging tecniques for mes adaptation wic fail to provide reasonable refined meses in case of discontinuous diffusion coefficients or te case of a constant diffusion coefficient tese estimators are based on a reference solution obtained by an averaging procedure of te numerical solution Let G u (V ) d defined by (372) G u (x) := An error indicator can ten be defined as meas d ( ) meas ω d (ω x ) k u, x N x (373) η 2 ZZ, := 1 k G u k u 2 L 2 ( ) In case of k 1 we obtain te well-known ZZ-estimator But in case of a discontinuous diffusion coefficient η ZZ, cannot serve as indicator is is clear if one observes tat te derivatives of te numerical solution tangential to te interface are continuous wereas te derivatives normal to te interface are not continuous but close to be continuous wen weigted wit k us in a proper averaging procedure te tangential and te normal part sould be weigted differently is may be ard to realize near singular points x, tat means on a part of te interface Γ 0 wic is not a straigt line e direction normal to te interface is not constant in Γ 0 ω x and tis will cause serious difficulties for properly defining te normal and tangential derivatives

23 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD Extension to more general problems In tis section we extend te estimators η R,η D to diffusion problems wit a mass term and boundary conditions of Diriclet-, Neumann- or Caucy-type in polygonal (polyedral) domains Ω R d,d =2, 3 As te mass term can become very big wen compared to te diffusion coefficient k tese problem comprise te so-called singularly perturbed reaction-diffusion problems Let a function g D H 1/2 (Γ D ) be given tat corresponds to te Diriclet boundary data and could be extended to g D H 1 (Ω) urter te function g N L 2 (Γ N ) corresponds to Neumann boundary data and functions g C L 2 (Γ C ),γ L 2 (Γ C ),γ > 0 to Caucy boundary data Let m L (Ω), 0 <mbe given As in section 32 we define te space V := { v H 1 (Ω) : v ΓD =0 } Let use define te energy scalar product a( u, v ):= k u v + Ω Ω muv + γuv Γ C As before we do not explicitly denote tat integration is done over te space variable x e variational form of te problem under consideration is as follows: we seek u, satisfying u g D + V and (381) a( u, v ) = fv + Ω γg C v + Γ C g N v Γ N v V We introduce te energy norm: v 2 Ω := k1/2 v 2 H 1 (Ω ) + m 1/2 v 2 L 2 (Ω ) + γ 1/2 v 2 L 2 ( Ω Γ C ), and a norm tat coincides wit te energy norm in te case Γ C = v 2 b,ω := k1/2 v 2 H 1 (Ω ) + m 1/2 v 2 L 2 (Ω ), for measurable subdomains Ω Ω Denote by b(, ) Ω te scalar product for te norm b,ω or simplicity we assume meas d 1 (Γ D ) > 0 Using Riesz s teorem one can sow tat tere exists a unique solution of problem (381) Let V V be an inite Element space wit continuous and piecewise linear functions Let g D, V be an approximation of g D en te solution of te discrete problem u satisfies u g D, + V and (382) a( u,v )= fv + γg C v + g N v, v V Ω Γ C Γ N We assume tat te diffusion coefficient k and te mass term m are piecewise constant on simplices of urter we assume tat functions g D = g D, and tat g N, g C and γ are piecewise constant on faces of Ω Oterwise an additional a-priori error occurs wic is due to te approximation of te boundary data

24 62 38 EXENSION O MORE GENERAL PROBLEMS A-posteriori error estimators were derived for te case k =1, Γ C = in [59] and for te case k =1,m =0in [37] Similar error estimators ave been derived in te case of a constant diffusion coefficient and pure Diriclet boundary conditions in [2] We derive a-posteriori error estimators for te case of discontinuous diffusion coefficients and Caucy boundary conditions by adapting ideas from [59] [37] We assume tat te mass term m is piecewise constant on simplices from and does not vary too muc from simplex to simplex: or any neigbouring simplices, wit it olds m m, were m,m denotes te value of m on te simplices, In te following we will skip te index and simply write m 381 Notation We will frequently make use of te following terms defined for a simplex resp a face { } { } := min,m 1/2, α := 1 1/2 min,m 1/4 α or eac face we need a special sape function adapted to te energy norm and wit support in ω is function was introduced in [59] We sortly sketc its definition Denote by te reference simplex wit vertices given by te unit vectors e i,i = 1,, d and te point e d+1 := 0 R d Let be te face opposite to te vertex e d or a given number 0 < δ 1 we introduce an affine transformation Ψ δ :(x 1,, x d 1,x d ) (x 1,, x d 1,δx d ) and denote te barycentric coordinates of te image δ := Ψ( ) by λ i,δ,i =1,, d +1 Now define te sape function on te reference simplex by: k 1/4 φ,δ := d d λd+1,δ Π i=1,,d 1 λi,δ wit support in δ Let us coose a simplex ω By te affine transformation G : mapping onto we define φ,δ (x) := φ,δ (G 1 (x)) e function φ,δ will be defined on te oter simplex saring te face in te same way at means tat φ,δ and φ coincide on te face and in te case δ =1tey coincide everywere or δ<1te function φ,δ will ave smaller support Wit an appropriate value of δ = α 2 k 1 and for k =1te function φ,δ can be viewed to be a function wic energy Ω is close to te extension from to ω wit minimal energy Ω [2] urter we extend te definition of te jump term j to te case of boundary conditions of tird kind or a face denote by, te simplices from ω ( 1 u 2 k n J := u g N k n + k u n γ (g C u ) k u n ) if Ω k 1/4 if Γ N if Γ C In te following we denote as usual te L 2 -scalar product on a subdomain Ω Ω by (, ) Ω IfΩ =Ω, we write (, )

25 CHAPER 3 ADAPIVE INIE ELEMEN MEHOD Stability of te interpolation operator in mixed norms In tis section we assume tat Γ C = e aim of tis subsection is to sow tat te interpolation operator I L : V V as te following interpolation properties wit respect to te energy norm defined in section 38 (383) (384) u I C (u) L 2 ( ) α u b, ω u I C (u) L 2 ( ) α u b, ω under te restriction to quasi-monotone diffusion coefficients Here ω and ω contains some neigbours of or or a definition of ω or ω see section 353 Similar estimates ave been sown for te case k =1[59] or te proof we use a trace inequality [59] Lemma 39 Let R d,d=2, 3 be a simplex wit diameter Let a face of en for u H 1 ( ) it olds PROO See [59] u 2 L 2 ( ) 1 u 2 L 2 ( ) + u L 2 ( ) u H 1 ( ) Here we state te interpolation results: Lemma 310 Let Γ C = and let te distribution of weigts k, be quasi-monotone or eac simplex and for eac face te bounds (383), (384) old PROO or te proof we exploit Lemma 39 and Lemma 33 A combination of (353), (354) yields u I L (u) 2 L 2 ( ) m 1 u 2 b, ω u I L (u) 2 L 2 ( ) 2 k u 2 b, ω proves (383) Coose a face and te neigboring simplex roug inequalities (383), (355) and Lemma 39 and we establis te bound u I L (u) 2 L 2 ( ) 1 u I L (u) 2 L 2 ( ) It remains to sow + u I L (u) L 2 ( ) u I L (u) H 1 ( ) 1 α 2 u 2 b, ω + α u b, ω 1 ( = 1 α α (385) 1 α α α 2 ) u 2 b, ω u kh 1 ( ω )