Mathematica Modeing and Numerica Anaysis ESAIM: MAN Modéisation Mathématique et Anayse Numérique Vo. 35, N o 4, 001, pp. 647 673 THE MORTAR METHOD IN THE WAVELET CONTEXT Sivia Bertouzza 1 and Vaérie Perrier Abstract. This paper deas with the use of waveets in the framework of the Mortar method. We first review in an abstract framework the theory of the mortar method for non conforming domain decomposition, and point out some basic assumptions under which stabiity and convergence of such method can be proven. We study the appication of the mortar method in the biorthogona waveet framework. In particuar we define suitabe mutipier spaces for imposing weak continuity. Unike in the cassica mortar method, such mutipier spaces are not a subset of the space of traces of interior functions, but rather of their duas. For the resuting method, we provide with an error estimate, which is optima in the geometricay conforming case. Mathematics Subect Cassification. 65N55, 4C40, 65N30, 65N15. Received: March 30, 000. Revised: January 3, 001. 1. Introduction In the ast few years there has been an increasing interest in the use of waveet based methods for the numerica soution of partia differentia equations. The existence of diagona preconditioners for eiptic operators of any given order [4, 8], as we as the possibiity of designing efficient adaptive approximation schemes for different type of probems [9,11,17,,9], are among the attractive features of such bases, which are raising the attention of the scientific computing community on the deveopment in this fied. On the other hand, in order for such methods to be appicabe in rea ife probems, severa issues sti need to be faced, among which the efficient treatment of non trivia geometries. We reca that such bases were first introduced in L (R) [31]. Generaizations to L (]0, 1[) (and to L (]0, 1[ d )) have been introduced in the eary nineties [,19], and recenty bases were constructed by a conforming domain decomposition approach, aowing to treat a domains which can be spit as union of conforma images of rectanges or cubes [15, 6]. Rather than foowing such a phiosophy, we prefer to consider here a non conforming domain decomposition approach. This has the great advantage of aowing to coupe discretizations of different types. By this approach, waveet bases can be couped for instance, with finite eements. This woud aow in principe to treat even very compicated geometries, by decomposing any given domain into subdomains, some of which are (conforma images of) squares or cubes and can then be discretized by waveets whie the remaining ones can be discretized with finite eements. Keywords and phrases. Domain decomposition, mortar method, waveet approximation. This work was partiay supported by the EC-TMR Network Waveets in Numerica Simuation, Contract No. ERB-FMRX- CT98-0184. It was done party whie the first author was a member of the Laboratoire ASCI (Université Paris Sud, Orsay). 1 I.A.N.-C.N.R., v. Ferrata 1, 7100, Pavia, Itay. e-mai: aivis@ian.pv.cnr.it LaboratoiredeModéisation et Cacu de IMAG, BP 53, 38041 Grenobe Cedex 9, France. e-mai: Vaerie.Perrier@imag.fr c EDP Sciences, SMAI 001
648 S. BERTOLUZZA AND V. PERRIER In particuar we consider here the mortar method [7, 8] which has been designed for such purpose. In the mortar method the interface of the decomposition is itsef decomposed into mortars (which in this paper wi be caed mutipier edges or sides ): each mortar is a whoe edge of a given subdomain, and the mortars are disoint from each other. Weak continuity is imposed by requiring that on each mortar the ump of the approximate soution is orthogona to a suitabe mutipier space. Such method has nowadays been appied to a wide variety of rea-ife probems in both two and three dimensions [3,4,33] and it is we suited for parae impementation [1]. We consider here the second version of such method [5], in which strong continuity of the discrete functions at cross points is not required. The aim of this paper is to introduce and anayze the appication of such a method in the context of biorthogona waveet bases. We imit our anaysis to the two dimensiona case, and for simpicity we assume that a the subdomains are rectanguar. For the sake of notationa simpicity we chose to spit the paper in two parts. In the first part we briefy review the theory of the mortar method. Whie doing that we underine the basic requirements on the discretization spaces needed in order to get stabiity and convergence. Moreover we point out the main differences between what happens in the framework of (for instance) finite eements and what wi happen when deaing with waveets. In the second part we show that, for a suitabe choice of the mutipier space, waveets fufi such requirements. Due to the particuar structure of the mortar method we are abe to do that by concentrating on one subdomain and one mortar. In particuar we provide an error estimate, which, in the case of a geometricay conforming decomposition, is optima.. The mortar method Rather than introducing and anayzing the mortar waveet method directy, for the sake of notationa simpicity we prefer to review at first the theory of the Mortar method in an abstract framework. In doing that, we wi point out some basic assumptions, which we wi ater check in the waveet case. We foow essentiay the guideines of [7, 8]. Most of the proofs are the same in an abstract framework as they are in the finite eement framework, and we wi therefore omit them, or sometimes ust briefy sketch them, for the sake of competeness. On the other hand, when deaing with waveets, one is naturay confronted with two new issues: the natura choice of the mutipier space on an edge is not necessariy a subset of the space of traces of the interior functions, as it is in the cassica mortar method; we wi see that, under suitabe assumptions which are easiy verified in the waveet case, this does not yied any maor modification in the resuts that can be obtained, with no substantia difference with respect to the cassica mortar method (in this respect see aso [36]); since interpoation on genera non dyadic (even coarse) grids in the waveet context is sti an open probem, we are ed to anayze the approximation error aso in the case in which the existence of an approximation operator that is interpoating at cross-points is not verified; we wi see that this eads (in the geometricay non conforming case) to a oss of a factor of the order of og(h) in the error estimate..1. The geometry Let Ω R be a poygona domain. We wi consider a decomposition of Ω as the union of L subdomains Ω, Ω= which, for the sake of simpicity we wi assume to be rectanguar. We set L Ω, (1) =1 Γ n = Ω n Ω, () S = Γ n. (3)
THE MORTAR METHOD IN THE WAVELET CONTEXT 649 Ω 1 Γ 1 Γ 13 Γ 3 Ω 3 γ (4) 1 γ (1) 1 Ω 1 γ (3) 1 γ () 1 γ (1) γ (4) 3 γ (4) 3 Ω 3 γ (3) 3 Ω Γ 4 Γ 34 Ω 4 γ (1) Ω γ (3) γ (1) 4 γ () 3 γ (4) 4 Ω 4 γ (3) 4 γ () γ () 4 Figure 1. Geometrica decomposition of the domain Ω. Throughout this section we wi assume that the above decomposition is fixed. The constants in the bounds that we are going to prove wi in genera depend on the size of the subdomains, uness otherwise stated. We denote by γ (i) (i =1,...,4) the i-th side of the th subdomain: Ω = 4 i=1 γ (i). For =1,...,L et ν denote the outer unit norma to the subdomain Ω, and for u H 1 (Ω ), et u / ν denote the outer norma derivative. Remark.1. The decomposition is said to be geometricay conforming if each edge γ (i) coincides with Γ n (= Ω Ω n )forsomen. If the decomposition is not geometricay conforming, then each interior edge γ (i) wi be in genera spit as the union of severa segments Γ n : γ (i) = Γ n, (4) n I (i) where I (i) individuates the set of indices n for which the subdomain Ω n is adacent to the edge γ (i) subdomain Ω : of the I (i) = {n : Ω n γ (i) 0} (5) We assume that the subdomains are reguar in shape, and from now on we wi make the foowing grading assumption (which is automaticay fufied in the case of a geometricay conforming reguar decomposition): (G1): the foowing bound hods max (,i) ( γ (i) min (i) n I ) ρ. (6) Γ n The constants appearing in the estimates of the foowing sections wi in genera depend on the bound ρ. For defining the mortar method we start by choosing a spitting of the skeeton S as the disoint union of a certain number of subdomain sides γ (i), which we wi ca mutipier sides (in the usua terminoogy these are caed non mortars or save sides ). More precisey, we choose an index set I {1,...,L} {1,...,4}
650 S. BERTOLUZZA AND V. PERRIER Ω 1 Ω 1 Ω 3 Ω 3 Ω Ω Ω 4 Ω 4 Figure. Two possibe choices of mutipier sides. For the one on the eft we have (using the indexing of Fig. 1) I = {(, 3), (1, ), (1, 3), (3, )} and I = {(4, 1), (4, 4), (3, 1), (, 4)} whie for the one on the right I = {(1, ), (3, 1), (3, ), (4, 1)}. Remark that in a cases the set {1,...,L} {1,...,4}\I I individuates the edges which beong to the externa boundary Ω. such that S = (,i) I γ (i), ( 1,i 1 ), (,i ) I, ( 1,i 1 ) (,i ) γ (i1) 1 γ (i) =. (7) Furthermore we wi denote by I {1,,L} {1,, 4} the index-set corresponding to trace sides ( mortars or master sides in the usua terminoogy), which is defined in such a way that I I = and S = (,i) I γ (i). (γ (i) Corresponding to the spitting (7) we define a norma direction on the skeeton. More precisey for (, i) I mutipier side) we set ν = ν on γ (i). (8) Since S = (,i) I γ (i), this defines ν on S. For any u =(u ) =1,...,L L =1 H1 (Ω ), accordingy: on γ (i),(, i) I (γ (i) mutipier side) we set u ν = u u aong the skeeton S with the proper sign: on Γ n γ (i), with (, i) I (γ (i) u ν wi be defined. Moreover, we et [u] denote the ump of ν mutipier side), [u] Γn = u Γn u n Γn. (9) In order to simpify the notation in the foowing we wi sometimes make use of a muti-index m =(, i). We wi for instance write γ m,m I (respectivey m I )forγ (i), (, i) I (respectivey (, i) I ). In the foowing, etting ˆΩ be either one of the domains Ω, Ω, Ω, γ (i) and Γ n, we wi denote by s, ˆΩ (resp. s, ˆΩ) the norm (resp. seminorm) of the space H s (ˆΩ). Moreover, we wi aso make use of the norm of the space H 1/ 00 (ˆΩ), which we wi simpy denote by 1/ H 00 (ˆΩ).
.. The continuous probem THE MORTAR METHOD IN THE WAVELET CONTEXT 651 For simpicity we wi consider the foowing mode probem. Given f L (Ω), find u :Ω R such that { u = f, in Ω, u =0, on Ω. (10) Given the spitting of the domain Ω introduced in the previous section, we wi consider a non conforming domain decomposition method for the soution of such a probem. In order to do that et where H 1/ ( Ω ) is defined by X = L { u H 1 } (Ω ) u =0on Ω Ω, (11) =1 T = L =1 H 1/ ( Ω ), (1) H 1/ ( Ω )=H 1/ ( Ω ) if Ω Ω = and { } H 1/ ( Ω )= η H 1/ ( Ω ),η Ω Ω 0 H 1/ 00 ( Ω \ Ω), otherwise. The space H 1/ ( Ω ) wi be endowed with the norm 1/, Ω, and we wi denote by 1/, the norm of the corresponding dua space. Remark that in genera 1/, 1/, Ω. Remark.. Remark that by definition the eements of both X and T (and in the seque the eements of the discrete subspaces X δ and T δ ) satisfy an homogeneous boundary condition on Ω. On X we introduce the foowing broken norm and semi-norm: u X = ( L =1 ) 1 ( L ) 1 u 1,Ω, u X = u 1,Ω. (13) In the foowing it wi aso be convenient to introduce the foowing norm on T : Moreover, for λ L (S) we wi use the notation: Let now a composite biinear form a X : X X R be defined as foows: =1 ( L ) 1/ η T = η 1/, Ω. (14) =1 ( L 1/ λ 1/,S = λ 1/,). (15) a X (u, v) = =1 L =1 Ω u v. (16)
65 S. BERTOLUZZA AND V. PERRIER The biinear form a X is ceary not coercive on X. In order to obtain a we posed probem we wi then consider proper subspaces of X, consisting of functions satisfying a suitabe weak continuity constraint. More precisey, for any subspace M of L (S) et a constrained space X (M) be defined as foows: X (M) = { u X : S } [u]λ =0, λ M (17) We wi consider the foowing probem (depending on the choice of the mutipier space M): Probem.3 (P M ). Find u M X(M) such that for a v X(M) a X (u M,v)= fv. (18) In the foowing we wi consider mutipier spaces M satisfying the foowing assumption. (BP): There exists a constant C M > 0 such that the foowing broken Poincaré inequaity hods for a u X (M) : Ω u X C M u X. (19) The bound (19) is evidenty equivaent to the coercivity of the biinear form a X over X (M). By simpy observing that M 1 M impies X (M ) X(M 1 ) one gets the foowing we known resut (see for instance [7]). Theorem.4. Let M satisfy assumption (BP). Then for a M such that M M we have the foowing: the soution u M of probem P M exists and is unique; for u soution of (10) the foowing bound hods with constant C = C( M):.3. Mortar discrete probem Approximation spaces u u M X C inf λ M u ν λ 1/,S. (0) For each et now Vδ be a famiy of finite dimensiona subspaces of H1 (Ω ) C 0 ( Ω ), depending on a parameter δ = δ > 0 and satisfying an homogeneous boundary condition on Ω Ω. Set and, for each edge γ (i) We set T δ = V δ Ω, (1) of the subdomain Ω et { } T,i = η : η is the trace on γ (i) of some u Vδ () { } T,i 0 = η T,i : η = 0 at the vertices of γ (i) (3) X δ = L Vδ X, T δ = =1 L Tδ T. (4) =1 For each m =(, i) I et a finite dimensiona mutipier space Mδ m parameter δ: on γ m be given, aso depending on the M m δ L (γ m ), dim(m m δ )=dim(t 0 m). (5)
THE MORTAR METHOD IN THE WAVELET CONTEXT 653 We set: M δ = {η L (S), η γm M m δ m I} m I M m δ. (6) The constrained approximation and trace spaces X δ and T δ are then defined as foows: We can now introduce the foowing discrete probem: { } X δ = v δ X δ, [v δ ]λ =0, λ M δ X(M δ ), (7) S { } T δ = η T δ, [η]λ =0, λ M δ. (8) S Probem.5 (PD). Find u δ X δ such that for a v δ X δ a X (u δ,v δ )= Ω fv δ. (9) The foowing resut hods: Theorem.6. Assume that M = δ>0 M δ satisfies assumption (BP). Then for a δ>0, probem P δ admits a unique soution u δ which satisfies the foowing error estimate: with C constant depending on M. ( u u δ X C inf u v δ X + inf u v δ X δ λ M δ ν λ 1/,S ), (30).4. Stabiity In order to appy the resut of the previous sections, we need to choose the mutipier spaces Mδ m in such a way that M = δ M δ m I δmδ m satisfies assumption (BP). We woud ike to reca that assumption (BP) is much ess restrictive than it might seem at first sight. It is we known [7] that in the framework considered here, a sufficient condition for (BP) to hod is the foowing: for a m =(, i) I (γ (i) mutipier side), for any piecewise constant function g, g constant on each Γ n, n I (i) (we reca that γ m = Γ n I (i) n ), we have gλ =0, λ M m = δ Mδ m impies g =0. (31) γ m In fact, roughy speaking, any function for which X = 0, does necessariy take a constant vaue on each subdomain, and, if it beongs to the constrained space X ( M), by (31) such constant vaues agree. Since the functions in X ( M) vanish at the boundary, the function is then identicay zero. The vaidity of assumption (31) has been studied for severa types of discretizations. Severa sufficient conditions for it to hod are therefore known. We reca for instance the foowing resut [6] which can be appied in our framework. Proposition.7 (Sufficient condition I). If card{i (i) } Ñ (card{i(i) } being the number of subdomains adacent to γ (i) and Ñ being the number of poynomias exacty reproduced in M m ), then (31) hods.
654 S. BERTOLUZZA AND V. PERRIER It is aso easy to prove that the foowing condition hods. Proposition.8 (Sufficient condition II). A sufficient condition for M m to satisfy (31) is that n I (i) there β n M m such that supp β n Γ n, β n =1. (3) Γ n In fact, for g = c n on Γ n we have γ m gβ n = Γ n gβ n = c n,andifg is orthogona to a the functions in M m this impies c n =0. Remark.9. In the case of a geometricay conforming decomposition (for a m =(, k), γ m =Γ n, for some n) the above condition is aso necessary, which is in genera not true in the case of geometricay non conforming decompositions (see for instance the case of the mortar eement method in the framework of a spectra decomposition, in which case the eements of Mδ m are poynomia functions)..5. Approximation error In order to bound the right hand side of (30) we wi make the foowing assumptions on the spaces considered: (A1): m =(, i) I (γ (i) mutipier side), there exists a bounded proection operator π m : L (γ m ) T m, 0 such that for a η L (γ m )andforaλ Mδ m (η π m η)λ =0, (33) γ m and for a η H 1/ 00 (γ m) π m η 1/ H 00 (γm) η H 1/ 00 (γm); (34) (A): m =(, i) I (γ (i) mutipier side), there exists a discrete ifting R m : Tm 0 V δ η Tm 0 such that for a R m η =0on Ω \γ m, R m η = η on γ m, (35) and R m η 1,Ω η 1/ H 00 (γm); (36) (A3): for a m =(, i) I (γ (i) mutipier side), the foowing inverse inequaity hods: for a eements η Tm 0 and for a s, 0 s<1/ ithods η 1/ H hs 1/ 00 (γm) m η s,γm, (37) where h m is a discretization parameter acting as mesh size on γ m. Remark.10. It is we known [35] that for s<1/ the Soboev space H s (G) (G bounded domain) can be obtained by space interpoation both as [L (G),H 1 (G)] s and as [L (G),H 1 0 (G)] s, the two resuting norms being equivaent. However, since the constants in the norm equivaence expode as s tends to 1/, it is not difficut to reaize that for (37) to hod uniformy in s, H s (γ m ) has to be defined as [L (γ m ),H 1 0 (γ m)] s. In fact, such inverse inequaities are usuay proven for s = 0 and then extended to s (0, 1/) by space interpoation. If
THE MORTAR METHOD IN THE WAVELET CONTEXT 655 then H s (G) is defined as [L (G),H 1 0 (G)] s, it is possibe to prove that for G = n G n, G and G n intervas, it hods for a u H 1/ ε (G) u 1/ ε,g n 1 ε u 1/ ε,g. (38) n Moreover we remark that, aso by space interpoation, assumption (A1) impies that the proection operator π m verifies for a s, 0<s<1/: uniformy in s. Letting the foowing emma hods: π m η s,γm η s,γm (39) h =min m I h m Lemma.11. If assumptions (A1 A3) hod, then for any η =(η ) =1,...,L T it hods: π m ([η]) ( 1+ og h ) H η 1/ 00 (γm) T. (40) m I Proof. Let η =(η ) =1,...,L be any eement of T = =1,...,L H1/ ( Ω ). Using (37) and (39), for any ε, 0 <ε<1/ ithods: m I π m ([η]) H 1/ 00 (γm) m I = m I m=(,i) L =1 hm ε π m ([η]) 1/ ε,γ m h ε h ε m ( η 1/ ε,γ n + η n 1/ ε,γ n ) n I (i) n:γ n η 1/ ε,γ n where the ast bound derives from (38). Choosing ε = 1 og h we get (40). h ε ε L η 1/, Ω We can now define a inear operator π : L =1 L ( Ω ) L =1 L ( Ω ) that we wi need in the foowing: more precisey, for η =(η ) =1,...,L, π(η) is defined on mutipier sides as π m appied to the ump of η, whie it is set identicay zero on trace sides and on the externa boundary Ω: π(η) =(η ) =1,...,L, with (41) η γ m = π m ([η] γm ), for m =(, i) I (4) η γm = 0, for m =(, i) I, η 0on Ω Ω. =1
656 S. BERTOLUZZA AND V. PERRIER By observing that for a η H 1/ ( Ω ) satisfying η γ (i) H 1/ 00 (γ(i) ), (i =1,...,4) it hods η 1/, Ω 4 i=1 η H 1/ 00 (γ(i) ) (43) we obtain the foowing coroary of Lemma.11. Coroary.1. If assumptions (A1 A3) hod, then for any η =(η ) =1,...,L in the trace space T it hods: We are then abe to prove the foowing theorem. π(η) T ( 1+ og h ) η T (44) Theorem.13. Let assumptions (A1 A3) hod. Then for any u H0 1 (Ω) we have: inf u v δ X ( 1+ og h ) ( L v δ X δ inf v =1 δ, Vδ u v δ, 1,Ω ) 1/. (45) Proof. For each et w δ, be an arbitrary eement of V δ. Let w δ =(w δ, ) =1,...,L. Since w δ does not necessariy satisfy the ump condition, it may not beong to X δ. We now define an eement w δ X δ as foows: w δ = w δ m I R m π m ([ w δ ]), (46) where, by abuse of notation for η T 0 m (m =(, i)) we indicate by R m (η) the eement of X δ which coincides with R m (η) inω and which is identicay zero on the other subdomains. We easiy check that w δ beongs to X δ. In fact for η T 0 m we have, and hence which impies, for λ δ M δ, thanks to (33), We can now bound [R m η]=η on γ m, [w δ ]=[ w δ ] π m ([ w δ ]) on γ m, S[w δ ]λ δ = m I γ m ( [ wδ ] π m ([ w δ ]) ) λ δ =0. m I R m π m ([ w δ ]) X = L =1 m I m=(,i) R m π m ([ w δ ]) 1,Ω L =1 m I m=(,i) R m π m ([ w δ ]) 1,Ω π m ([ w δ ]) = π H 1/ 00 (γm) m ([ w δ u]) H 1/ (γm), 00 m I m I
THE MORTAR METHOD IN THE WAVELET CONTEXT 657 where the ast equaity descends from the observation that for u H 1 (Ω)ithods[u] =0, and hence [ w δ ]= [ w δ u]. Thus, by appying emma.11 for η =[ w δ u], using a cassica trace theorem we obtain inf v δ X δ u v δ X u w δ X + m I R m π m ([ w δ ]) X ( 1+ og h ) L u w δ, 1,Ω. Since the w δ, are arbitrary, we get the thesis. Coroary.14. Let the famiy Vδ be given and et M δ be defined by (6) and X δ by (7). Assume that the foowing properties hod: (i) accuracy: assumptions (A1 A3) are satisfied for Tm 0 and M δ m defined by (3), (5); (ii) stabiity: M = δ M δ satisfies assumption (BP). Then, for u soution of (10) and u δ soution of (9) the foowing error estimate hods: =1 u u δ X ( 1+ og h ) ( L inf v =1 δ, Vδ + inf λ M δ u ν λ 1/,S. u v δ, 1,Ω ) 1/ (47) Remark.15. We remark that assumptions (A1 A3) dea independenty with each subdomain and aso with the couping between the discretization on a subdomain and the mutipier space Mδ m defined on each one of its sides. In the construction of suitabe discretization spaces, it wi then be sufficient to study the properties of the discretization on one subdomain and the mutipier space induced on one of its sides. Ceary, Theorem.13 yieds ony a sub-optima error estimate, where, due to the constraint, a factor of the order og h is ost with respect to the optima approximation rate. Nevertheess, if the soution is sufficienty reguar, an optima error estimate can be retrieved, provided that a suitabe proector exists, verifying an interpoation property at cross points. More precisey the foowing theorem hods. Theorem.16. For some s >1, assume that Vδ H s (Ω ) and that operators Π δ, : H s (Ω ) Vδ exist such that s, t, 0 s s, s t t, ( t possiby depending on ) for a u H t (Ω ) with u =0on Ω Ω we have δ being the mesh-size of the discretization in the subdomain Ω ; for a A Ω such that A is a vertex of Γ n for some n, u Π δ, u s,ω δ t s u t,ω, (48) Π δ, u(a) =u(a), u H s (Ω ). (49) Then if u H s (Ω) H 1 0 (Ω) satisfies u Ω H t (Ω ) ( s t t) it hods inf u v δ X v δ X δ L =1 δ t 1 u t,ω. (50) Remark.17. We point out that in the case of genera discretizations, for m =(, i) I (γ (i) mutipier side) the two mesh size parameters h m in the inverse inequaity (37) (corresponding to the finest mesh size on γ m ) and δ in the direct inequaity (48) (corresponding to the coarsest mesh size in Ω ) do not necessariy coincide.
658 S. BERTOLUZZA AND V. PERRIER Proof. Let w δ, =Π δ, u. Setting w δ =(w δ, ) =1,...,L and w δ = w δ m I R mπ m ([ w δ ]), as in the proof of Theorem.13 we have inf u v δ X u w δ X v δ X δ L =1 u Π δ, u 1,Ω + m I Thanks to (49), [w δ u] beongs to H 1/ 00 (γ m), hence we can write π m ([ w δ u]) 1/ H 00 (γm) [ w δ u] 1/ H (γm). 00 m I m I π m ([ w δ u]) H 1/ 00 (γm). We now observe that [ w δ u] Γn H 1/ 00 (Γ n) for a, n. Then, by writing it as the sum of functions ζ n each one coinciding with [ w δ u] onγ n and vanishing identicay on γ m \ Γ n, since the zero extension operator is bounded from H 1/ 00 to H 1/ 00 we get [ w δ u] 1/ H 00 (γm) n I (i) ζ n 1/ H [ w 00 (γm) δ u] 1/ H n I (i) 00 (Γ n) whence, since on Γ n, [ w δ u] Π δ, u u + Π δ,n u u π m ([ w δ u]) 1/ H 00 (γm) Π δ, u u 1/ H 00 (Γ n). m I,n The concusion foows by observing that Π δ, u u H 1/ 00 (Γ n) δt 1 u t,ω. This ast bound can be proven by space interpoation [35]. On one hand, we have Π δ, u u H s 1/ 0 (Γ n )and On the other hand Π δ, u u s 1/,Γn Π δ, u u s 1/, Ω (51) Π δ, u u s,ω δ t s u t,ω. (5) Π δ, u u 1/4,Γn Π δ, u u 3/4,Ω δ t 3/4 u t,ω. Since H 1/ 00 (Γ n) can be obtained as the interpoated of order θ =1/[4( s 1) + 1] between H 1/4 (Γ n )and H s 1/ 0 (Γ n ), we obtain that Π δ, u u H 1/ 00 (Γ n) δ(1 θ)(t 3/4) δ θ(t s) u t,ω = δ t 1 u t,ω. Coroary.18. Under the assumptions of Theorem.16, if for a =1,...,L the soution u of (10) satisfies u Ω H t (Ω ) for some t, s t t then u u δ X inf u λ M δ ν λ 1/,S + L =1 δ t 1 u t,ω. (53)
THE MORTAR METHOD IN THE WAVELET CONTEXT 659 Remark.19. If we consider the Lagrange mutipier formuation of probem (9), Coroary.1 woud impy that a discrete inf-sup condition of the form inf sup λ δ M δ u δ X δ S [u δ]λ δ λ δ 1/,S u δ X α(h) > 0 is fufied with a stabiity constant α = α(h), decreasing as og h 1. In fact, it is easy to see that Fortin s Lemma [13] can be appied by setting the Fortin proector equas to (Id π). An aternative to such an approach is to work with suitabe mesh dependent norms. Then, in the finite eement framework, a discrete inf-sup condition can be proven to hod uniformy in h [1]. However, working with mesh dependent norms usuay yieds (when, as in the present framework it is not possibe to work triange by triange ), a dependence of the constant in the estimates obtained on the ratio between the coarser and the finer mesh sizes of each subdomain. This is due to the concurrent use of direct and inverse inequaities. In the case of a very non uniform discretization, as one woud have in an adaptive waveet scheme, such a ratio can be much arger than the ogarithmic factor in the bound (44) and in the resuting error estimate. Therefore, though for the sake of simpicity we wi concentrate, ater on, on a uniform waveet discretization, we chose here to use natura norms such as 1/,S. We remark that in the geometricay conforming case, if the soution is sufficienty reguar, we sti get an optima error estimate (see Th..16 and 3.1). 3. The mortar waveet method We now come to the probem of constructing mortar approximation spaces in a genera waveet context. In view of Remark.15 we focus here on one (rectanguar) subdomain Ω which for simpicity we identify with the unit square. The approximation spaces on ]0, 1[ wi be obtained from tensor-product of one-dimensiona spaces. Starting from a (now cassica) mutiresoution anaysis on the interva (see for instance [, 19]), we construct a waveet famiy (adapted to the mutipier sides) which wi aow to define a suitabe mutipier space verifying by construction the basic assumptions (A.1 A.3) needed to appy the abstract resut of the previous section. 3.1. Mutiresoution anayses on the interva and approximation properties 3.1.1. Scaing functions on the interva The construction of mutiresoution anayses and associated waveet bases on the interva, which preserve the approximation properties of the waveet bases on R has nowadays aready been discussed in a number of papers (see [, 19] for the first constructions, but aso [18, 5, 30, 34]). To be as genera as possibe, we wi consider the case of biorthogona waveet bases, that incudes the one of orthonorma bases. Let us point out which are the properties of such bases, which wi be needed for the design and the anaysis of the mortar waveet method and which we wi assume to be verified by the chosen basis. We reca that a wide cass of bases exists, which satisfy by construction such assumptions. We assume that we are given a coupe of biorthogona muti-resoution anayses (MRA) of L (0, 1), that is a coupe of increasing sequences of finite dimensiona subspaces (V = V (]0, 1[)) 0 and (Ṽ = 1[)) Ṽ(]0,, 0 whose respective union is dense in L (0, 1). Without oss of generaity, we can assume that dimv = dimṽ = + (see [, 1, 3] for exampe). The spaces V and Ṽ are respectivey spanned by biorthogona scaing function Riesz bases (ϕ,k ),..., and ( ϕ,k ),...,, verifying: 1 0 ϕ,k ϕ,k = δ kk, k, k =0,...,.
660 S. BERTOLUZZA AND V. PERRIER We reca that by the definition of Riesz s basis, the two foowing norm equivaences hod uniformy in : u k ϕ,k 0,]0,1[ u k 1/ (54) and u k ϕ,k 0,]0,1[ u k 1/. (55) The scaing function bases are usuay constructed as a modification of the corresponding (compacty supported) scaing functions in L (R) (see [7]). In this process, one naturay distinguishes between edge (eft and right) functions and interior functions: interior functions coincide with scaing functions on the ine whose support is incuded into ]0, 1[ whie edge function are inear combinations of scaing functions on the ine (restricted to ]0, 1[), whose support overaps the eft (resp. right) edge. Consequenty, for 0, the scaing basis of V is usuay indexed as foows, with N a given integer: ϕ eft,k,,...,n 1, the N scaing functions at the eft edge; ϕ,k,k= N,..., N + 1, the interior scaing functions; ϕ right,k,k= N +,...,,then scaing functions at the right edge. Simiary for the scaing basis of Ṽ biorthogona (again Ñ is a given integer): ϕ eft,k,,...,ñ 1, the Ñ scaing functions at the eft edge; ϕ,k,k= Ñ,..., Ñ + 1, the interior scaing functions; ϕ right,k,k= Ñ +,...,,theñ scaing functions at the right edge. For notationa simpicity, we wi omit in the foowing the suffixes eft and right. We wi then denote (ϕ,k ),..., and ( ϕ,k ),..., the above bases. We reca that these functions have compact support and are scae invariant, i.e. 0, x [0, 1], and k =0,...,N 1(resp. k =0,...,Ñ 1 for the second equaity) on the eft boundary it hods ϕ,k (x) = 0 ϕ 0,k( 0 x), ϕ,k (x) = 0 ϕ 0,k( 0 x), whie on the right boundary we have k = N +,..., (resp. k = Ñ +,...,) ϕ,k (1 x) = 0 ϕ 0,k( 0 (1 x)), ϕ,k (1 x) = 0 ϕ 0,k( 0 (1 x)). Moreover, the interior scaing functions coincide with the origina scaing functions on the rea ine and 0, k = N,..., N + 1, and they take the form: ϕ,k (x) =ϕ,n (x (k N)) = / ϕ( x k), where ϕ(x) = 0/ ϕ 0,N( 0 (x + N)) is the scaing function of the corresponding mutiscae anaysis for L (R). An anaogous reation hods for the duas ϕ,k. Finay we can aso suppose, that a scaing functions of V vanish at the edges 0 and 1, except one function at each edge. For exampe we wi assume that ony the functions ϕ,0 and ϕ, verify a non-vanishing boundary
THE MORTAR METHOD IN THE WAVELET CONTEXT 661 condition: ϕ,0 (0) 0 and ϕ, (1) 0, ϕ,k (0) = 0, k =1,...,, (56) ϕ,k (1) = 0, k =0,...,. We wi assume that the two MRA are respectivey R-reguar and R-reguar, that is for a, k in the range considered it hods ϕ,k H R (0, 1) and ϕ,k H R(0, 1), with R 1, R >0. Foowing [19], these scaing functions are constructed in such a way that they satisfy Strang-Fix conditions, that is they aow to reconstruct poynomias up to degree N 1inthespaceV and up to degree Ñ 1inthe space Ṽ. More precisey, we wi have for a 0 and for n =0,...,N 1 / ( x) n = a n k ϕ,k (x), and / ( (1 x) ) n = b n k ϕ,k (x), (57) and for n =0,...,Ñ 1 / ( x) n = ã n k ϕ,k(x), and / ( (1 x) ) n = bn k ϕ,k (x), (58) with a n k and ãn k reas independent of. We reca that the parameters R, R, N and Ñ necessariy satisfy the reations R N and R Ñ. Let now V 0 H1 0 (0, 1) be the space of functions of V vanishing at the boundaries of the interva: V 0 = V H0 1 (0, 1) = span ϕ,k, k =1,...,, V = V 0 span ϕ,0,ϕ,.. When designing the mortar waveet method, the space V 0 wi pay the roe of the space which, in Section.3, was denoted Tm 0, that is the space of traces on a mutipier side of discrete functions, vanishing at the extrema. The corresponding mutipier space (which wi pay the roe of the Mδ m of Sect..3) is most naturay defined with the aid of a suitabe dua space, which wi have to satisfy the assumptions of Section.5. The construction of such dua space for V 0 is the obect of the foowing theorem. Theorem 3.1. Let 0. Assume that the two inear systems Ñ 1 ã n k α k =ã ñ N, and bn k β k = b n Ñ+1, n =0, Ñ 1 (59) k= Ñ+
66 S. BERTOLUZZA AND V. PERRIER admit soutions (α k ), Ñ 1 and (β k) k= Ñ+, satisfying α 0 0and β 0. Then the famiy: ϕ,k = ϕ,k α k α 0 ϕ,0, for k =1, Ñ 1 ϕ,ñ = ϕ, Ñ + 1 α 0 ϕ,0, ϕ,k = ϕ,k, for k = Ñ +1, Ñ (60) ϕ, Ñ+1 = ϕ, Ñ+1 + 1 β ϕ,, ϕ,k = ϕ,k β k β ϕ,, for k = Ñ +, +1 is biorthogona to the basis { ϕ,k, k =1,..., } of V 0, that is for a n, k =1,..., it hods 1 0 ϕ,k ϕ,n = δ n,k. Moreover the space Ṽ defined by: contains a poynomias of degree Ñ 1. Remark that Ṽ Ṽ = span ϕ,k, k =1,..., (61) does not verify homogeneous boundary conditions. Proof. The biorthogonaity between famiies ( ϕ,k )and(ϕ,k) is a trivia consequence of the biorthogonaity between ( ϕ,k )and(ϕ,k ). By construction, the ( ϕ,k ) reproduce poynomias up to order Ñ 1. Indeed, by equation (58) we have, n =0,...,Ñ 1: / ( x) n = = ã n k ã n 0 + ϕ,k(x) Ñ 1 ã n α k k ãñ N ϕ,0 (x)+ α 0 α 0 Ñ ã n k ϕ,k(x)+ k= Ñ+1 ã n k ϕ,k (x) where we appied the definition (60). Since the α k satisfy the foowing equation: we have / ( x) n = Ñ 1 ã ñ N Ñ ã n kα k =0, n =0, Ñ 1 (6) ã n k ϕ,k (x)+ k= Ñ+1 ã n k ϕ,k(x).
THE MORTAR METHOD IN THE WAVELET CONTEXT 663 Then each poynomia P of degree Ñ writes: P (x) = Ñ p k ϕ,k(x)+ k= Ñ+1 p k ϕ,k (x) with p k =<P ϕ,k > for k =1,..., + 1. In particuar / ( (1 x) ) n Ñ = bn k ϕ,k (x)+ k= Ñ+1 bn k ϕ,k (x). (63) Using (60) again this yieds / ( (1 x) ) n Ñ = bn k ϕ,k (x)+ bn Ñ+1 + k= N+ bn k β k β bn Ñ+1 β ϕ, (x). Since the β k satisfy the foowing equation: we have bn Ñ+1 k= N+ bn k β k =0, n =0, Ñ 1 (64) / ( (1 x) ) n = bn k ϕ,k (x). (65) Since ceary the set { ( / (1 x) ) n, n =0,..., Ñ 1} generates the set of poynomias of degrees ess or equa than Ñ 1, this yieds the thesis. Remark 3.. We assumed here that the matrices [ã n k ] 0 k,n Ñ 1 and [ b n k ] +Ñ+ k,n are invertibe and that the coefficients α 0 and β 0 are non vanishing. This can be proven in particuar cases, such as for biorthogona spine waveets [0, 5] or orthogona waveets. In this ast exampe ϕ,k = ϕ,k and for instance on the eft boundary the coefficients ã n k = 1 0 xn ϕ,k can be written as ã n k = p n(k), where p n is a poynomia of degree n [3]. [ã n k ] 0 k,n Ñ 1 is then a nonsinguar Vandermonde type matrix. The fact that the coefficients α 0 and β are non vanishing descends from the same argument, by considering the matrix [ã n k ] 0 n Ñ 1,1 k Ñ. In genera we wi have to verify case by case that such an assumption hods. Remark that, thanks to the scae invariance property of the scaing functions, it wi be enough to verify it once and for a for = 0. 3.1.. Proectors on MRA spaces and approximation properties Let P and P be the biorthogona proectors associated to V and Ṽ defined, as usua, for a η and λ in L (0, 1) by: P (η) = η ϕ,k ϕ,k P (λ) = λ ϕ,k ϕ,k (66)
664 S. BERTOLUZZA AND V. PERRIER where denotes the L (0, 1) scaar product. It is we known that under the assumptions of the previous section, the foowing Jackson and Bernstein inequaities hod: Theorem 3.3. Let 0 r s N, r R, then for a η H s (0, 1), we have: whie, for 0 r s Ñ, r R and for a λ H s (0, 1) it hods: Moreover, a scaing argument yieds the foowing inverse inequaities Theorem 3.4. For a r, s, 0 r s R one has: and for a r, s, 0 r s R one has: η P (η) r,]0,1[ (s r) η s,]0,1[, (67) λ P (λ) r,]0,1[ (s r) λ s,]0,1[. (68) η s,]0,1[ (s r) η r,]0,1[, if η V, (69) λ s,]0,1[ (s r) λ r,]0,1[, if λ Ṽ. (70) In the foowing it wi aso be usefu to consider the biorthogona proector on V 0 its adoint π. Definition 3.5. Let π : L (0, 1) V 0 be defined, for a η L (0, 1) by: Moreover et π : L (0, 1) Ṽ be the adoint of π : induced by its dua Ṽ and π (η) = η ϕ,k ϕ,k. (71) π (λ) = λ ϕ,k ϕ,k. (7) The natura environment for anayzing the proectors π consists in the spaces with nu zero-th order trace. More precisey, for s 1 we denote by H0 s (0, 1) the space H s 0(0, 1) = H s (0, 1) H 1 0 (0, 1), endowed with the H s norm. For s =0wesetH0 0(0, 1) = L (0, 1), and for 0 <s<1 we define H0 s by space interpoation. We remark that H 1/ 0 (0, 1) = H 1/ 00 (0, 1). Since V 0 V and Ṽ Ṽ and since they reproduce ocay respectivey a poynomias of degree N 1 satisfying homogeneous boundary condition at the edge 0 and 1 and poynomias of degree Ñ 1, the foowing bounds aso hod. Theorem 3.6. For a r R, for a s, 0 r s N, π is continuous from H0 s(0, 1) to Hr 0 (0, 1), and for a η H0 s (0, 1) we have: η π (η) H r 0 (0,1) (s r) η H s 0 (0,1). (73)
THE MORTAR METHOD IN THE WAVELET CONTEXT 665 For a r R, for a s, 0 r s Ñ, for a λ Hs (0, 1), we have: Moreover, for a 0 r s R, for a η V 0, it hods λ π (λ) r,]0,1[ (s r) λ s,]0,1[ (74) η H s 0 (0,1) (s r) η H r 0 (0,1). (75) Proof. It is not difficut to check that both π and π are L (0, 1) bounded proectors. Let us for exampe consider π (η): using the definition of ϕ,k and the Riesz s basis property (54) it is not difficut to reaize that π (η) 0,]0,1[ = η ϕ,k ϕ,k 0,]0,1[ η ϕ,k P (η) 0,]0,1[ η 0,]0,1[ As far as π is concerned, we first observe that, by the definition of ϕ,k (54) it hods and using the Riesz s basis property u k ϕ,k 0,]0,1[ Ñ u k + c k u k + where the coefficients c k and d k are given by (60): k= Ñ d k u k c k = α k,,... α,ñ, c 0 = 1, d k = β k, k = 0 α 0 β Ñ +,...,, d Ñ+1 = 1 β Then u k, π (λ) 0,]0,1[ = λ ϕ,k ϕ,k 0,]0,1[ λ ϕ,k P (λ) 0,]0,1[ λ 0,]0,1[. Using the poynomia reproduction properties of the spaces V 0 and Ṽ, as we as the Bernstein inequaities (69) and (70), the theorem then resuts by appying standard arguments. In particuar it hods Coroary 3.7. π is continuous from H 1/ 00 Moreover, for a η L (0, 1) and for a λ Ṽ, 3.. Waveets (0, 1) to H1/ 00 π (η) 1/ H 00 (0,1) η H 1/ 00 1 0 (0, 1), that is for a η H1/ 00 (0, 1): (0,1). (76) (η π (η))λ =0. (77) We can now introduce two coupes of biorthogona waveet bases, which wi both be needed for the anaysis and/or for the impementation of the mortar waveet method. On one hand we wi need the compement spaces W and W defined as foows.
666 S. BERTOLUZZA AND V. PERRIER Definition 3.8. The compement spaces W and its dua W are defined by: Let us now introduce the compement space W 0 of V 0 in V+1 0 biorthogona proector π defined in the previous section. W =(P +1 P )V +1, (78) W =( P +1 P )Ṽ+1. Definition 3.9. The compement spaces W 0 and its dua W are defined by: and its dua. To do this, we wi use the W 0 =(π +1 π )V 0 +1, (79) W =( π +1 π )Ṽ +1. Foowing [0] it is possibe to construct waveet bases for W, W, W 0 and W. We wi then have biorthogona waveet Riesz bases {ψ,k,,, }, { ψ,k,,, },and{ψ,k 0,,, }, { ψ,k,,, } such that the foowing identities hod for a f L (0, 1): (P +1 P )f = f ψ,k ψ,k, (80) ( P +1 P )f = f ψ,k ψ,k, (81) (π +1 π )f = f ψ,k ψ0,k, (8) ( π +1 π )f = f ψ,k 0 ψ,k. (83) Remark 3.10. The functions in the compement spaces W do not satisfy an homogeneous boundary condition. As usua in the waveet framework, norm equivaences hod for Soboev spaces of negative and/or fractiona smoothness (either of the type H s (0, 1) or H0(0, s 1) as defined above), in terms the expansions in either one of the bases {ψ,k }, { ψ,k }, {ψ,k 0 } and { ψ,k }. It is beyond the goas of this paper to precisey state such norm equivaences. We wi therefore imit ourseves to state the foowing theorem, which wi pay a key roe in both the anaysis and the impementation of the Mortar Waveet Method and which can be proven by appying the resuts of [14, 3]. Theorem 3.11. Let f L (0, 1). The foowing two norm equivaences hod: f H 1/ (0, 1), f 1/,]0,1[ 0 +1 f ϕ 0,k + 0 f ψ,k, (84) 0 f H 1/ 00 (0, 1), f f ϕ H 1/ 00 (0,1) 0,k + f ψ,k. (85) 0 Remark 3.1. Moreover f H 1/ (0, 1) (resp. f H 1/ 00 (0, 1)) if and ony if the sum on the right hand side of (84) (resp. (85)) is finite.
THE MORTAR METHOD IN THE WAVELET CONTEXT 667 3.3. Mortar approximation spaces in the waveet framework 3.3.1. D mutiresoution anayses As usua in the unit ]0, 1[ we wi consider as approximation spaces a two-dimensiona MRA V defined by tensor products of one-dimensiona MRA: V = V V span Φ,k, k K, Ṽ = Ṽ Ṽ span Φ,k, k K, where the muti-index set K is defined by K = {k =(k 1,k ), k 1,k =0,..., } = {0,..., } and where, using the notation (f g)(x, y) =f(x)g(y) the functions Φ,k are defined by Φ,k = ϕ,k1 ϕ,k and Φ,k = ϕ,k1 ϕ,k. In the same way, V 0, the subspaces of V verifying homogeneous boundary conditions and its dua Ṽ K = {1,..., } K are, for V 0 = V 0 V 0 span Φ,k, k K, Ṽ = Ṽ Ṽ span Φ,k, k K. It is we known that the famiy (V ) constitutes a MRA of L (]0, 1[ ), and (V 0)aMRA of H1 0 (]0, 1[ ). The two-dimensiona biorthogona proections on respectivey V and V 0 wi be denoted by P and Π and their adoint by P and Π. They are defined respectivey as: P f = f Φ,k Φ,k, Π f = k K P f = f Φ,k Φ,k, Π f = k K f Φ,k Φ,k, k K f Φ,k Φ,k. k K Two-dimensiona waveets are constructed (as usua) by tensor products of univariate bases. In particuar, we wi then have two coupes of biorthogona waveet bases, which we wi denote by {Ψ,κ } κ I, { Ψ,κ } κ I,and {Ψ 0,κ } κ I, { Ψ,κ } κ I defined in such a way that: (P +1 P )f = κ I f Ψ,κ Ψ,κ (86) ( P +1 P )f = κ I f Ψ,κ Ψ,κ (87) (Π +1 Π )f = κ I f Ψ,κ Ψ0,κ (88) ( Π +1 Π )f = f Ψ 0,κ Ψ,κ, (89) κ I
668 S. BERTOLUZZA AND V. PERRIER where I and I wi denote suitabe muti-index sets. The corresponding compement spaces wi then be denoted by W = (P +1 P )V = span Ψ,κ, κ I W = ( P +1 P )V = span Ψ,κ, κ I W 0 = (Π +1 Π )V = span Ψ 0,κ, κ I W = ( Π +1 Π )V = span Ψ,κ, κ I (90) Remark 3.13. There are severa ways of buiding waveet bases for the spaces W, W, W 0 and W. The basis which is cassicay used in the context of waveet discretization is constructed starting from the observation that the space W can be decomposed as W =(V W ) (W V ) (W W ). The functions in the basis take then the three forms ϕ,k ψ,k, ψ,k ϕ,k and ψ,k ψ,k. A second basis for W can be obtained by further decomposing V as V = V 0 1 m= 0 W m, which yieds a decomposition for W of the form W =(V 0 W ) (W V 0 ) 1 m= 0 (W m W ) m= 0 (W W m ) The basis functions take then the forms ϕ 0,n ψ,k, ψ,k ϕ 0,n, ψ m,n ψ,k and ψ,k ψ m,n (with m ). The same aternative hods for the three other spaces W, W 0 and W. Depending on the choice made, the index sets I and I wi have different forms. In the first case, the muti-index κ wi be a tripet κ =(k, k,ɛ), where the type parameter ɛ =1,, 3 distinguishes between the three forms ϕ,k ψ,k, ψ,k ϕ,k and ψ,k ψ,k. In the second case, it wi be a quadrupet (m, n, m,n ) with suitabe restrictions on the vaues of m and m (max{m, m } = and, using the convention ψ 0 1,k = ϕ 0,k, m, m 0 1). We remark however that Theorem 3.15 in the foowing hods for both choices. Again, Jackson and Bernstein inequaities and norm equivaences anaogous to (67 73) and (84 85) are sti vaid in dimension. In particuar, the foowing two theorems hod. Theorem 3.14. For a r, 0 r R, for a s, r s N and for a u H s (]0, 1[ ) it hods u P (u) r,]0,1[ (s r) u s,]0,1[. (91) For a r, s, 0 r s R, and for a u V u s,]0,1[ (s r) u r,]0,1[. (9) Theorem 3.15. The foowing norm equivaences hod: for a u H 1 (]0, 1[ ) u 1,]0,1[ u Φ 0,k + u Ψ,κ, (93) k K 0 κ I 0 and for a u H 1 0 (]0, 1[ ) u 1,]0,1[ u Φ 0,k + u Ψ,κ. (94) k K 0 0 κ I
THE MORTAR METHOD IN THE WAVELET CONTEXT 669 3.3.. Trace and mutipier spaces In order to use such spaces in the framework of the mortar method, for each edge of ]0, 1[ we need to define a space suitaby couped with the space of traces of the functions in V, as required in Section.5, to be used as mutipier space if the edge is chosen to be a mutipier edge. Let then γ be any edge of ]0, 1[. Thanks to the tensor product structure of the space V,thespaceT (γ) of traces of functions of V can be identified to the space spanned by the basis of scaing functions on the interva: T (γ) =V γ V = span ϕ,k, k =0,...,. (95) Then, the space T 0 (γ) H1/ 00 (γ) of functions of T (γ) vanishing at the extrema of γ verifies: T 0 (γ) V 0 = span ϕ,k, k =1,...,. (96) The choice of the mutipier space on the edge γ is the obect of the foowing definition. Definition 3.16. If γ is a mutipier side a natura choice for the mutipier space M (γ) onγ wi be: M (γ) Ṽ = span ϕ,k,,...,. (97) Remark 3.17. The mutipier space M verifies M T, and not M T as usua in the mortar methods, see [8] (we fa back in the cassica mortar method framework, i.e.m is a subspace of codimension of T,if orthonorma waveets are considered). Such a choice has some advantage over a choice impying M T. Such a space is in fact exacty the dua space of T 0. Thanks to the biorthogonaity property of the two bases for the spaces M and T 0, the matrix appearing in the inear system that has to be soved for computing the proector π is diagona. This aso happens for spectra approximation, whie for P1 finite eements, the corresponding inear system invoves a tridiagona matrix. We reca that the computation of the proector π is needed for imposing the constraint in the numerica resoution of the inear system stemming from (9), if one wants to avoid the expicit construction of a basis for X δ. Theorem 3.18. The spaces T 0 (γ) and M (γ) verify assumptions (A.1 A.3). Proof. The proection: π : L (γ) T 0 (γ) defined by (71) verifies assumption (A1). This resut is given by Coroary 3.7. Moreover, if γ is a mutipier side of ]0, 1[, for a η T 0 and s< 1, we have by (75): η H 1/ 00 (γ) (s 1/) η s,γ, and the assumption (A.3) is fufied with h =. Then we ony need to prove that (A) hods. For simpicity et us assume that γ = {(x, 0), x ]0, 1[}. The foowing map: R : T 0(γ) V defined, for a η T 0 (γ) by: R (η) = 1 = 0 η ψ,k 0 ϕ,0(0) ψ0,k ϕ,0 + η ϕ 0,k ϕ 0,0(0) ϕ 0,k ϕ 0,0 is a ifting from H 1/ 00 (γ) toh1 (]0, 1[ ) which verifies assumption (A). In fact, et us consider the MRA ˇV = V 0 V and its dua ˆV = Ṽ Ṽ. If we denote by { ˇΨ,κ,κ Ǐ} and { ˆΨ,κ,κ Ǐ}, the corresponding coupe of biorthogona waveet bases, (with an anaogous notation for scaing functions) the foowing norm equivaence hods: for a u H0 1 (]0, 1[ ), satisfying u(0,y) u(1,y) 0 for a
670 S. BERTOLUZZA AND V. PERRIER y ]0, 1[: u 1,]0,1[ k Ǩ 0 u Φ 0,κ + f ˆΨ,κ, 0 κ Ǐ where Ǩ 0 = {1,..., 0 } {0,..., 0 +1}. Now we observe that for suitabe κ = κ(k) wehave ˇΨ,κ (x, y) =ψ 0,k(x)ϕ,0 (y), ˆΨ,κ (x, y) = ψ,k(x) ϕ,0 (y). Therefore, appying the previous norm equivaence to R (η), and observing that ϕ,0 (0) / we get 0 R (η) 1,]0,1[ 0 η ϕ 0,k + 1 = 0 η ψ,k, (98) and the right hand side of (98) is equivaent to η 1/ H 00 (γ), thanks to norm equivaence (85). 3.4. Error estimates We can now use the function spaces ust defined in the framework of the mortar method described in Section. For simpicity we wi ust consider the case in which a the subdomains are discretized by waveets. More precisey, for each =1,...,Let F :Ω ]0, 1[ be a inear mapping of the (rectanguar) subdomain Ω onto the reference domain and for each edge γ (i) et F (i) : γ (i) ]0, 1[ (F (i) on γ (i) by F. In each subdomain Ω et the approximation space Vδ be given by: V δ = V () F with some fixed () 0.Toeachedgeγ (i) we associate the trace space of V δ : For m =(, i) I (γ m = γ (i) T,i = T () F (i) mutipier side), the mutipier space is F γ (i) ) be the mapping induced M m δ = M () F (i). The constrained approximation space X δ and its mutipier space M δ are then defined according to (7, 6). If the decomposition is geometricay conforming, the resuting mutipier space M δ wi automaticay satisfy assumption (31) (ensuring stabiity), whie in the case of a geometricay non conforming decomposition we wi have to choose () in such a way that either one of the sufficient conditions I and II of Section.4 are satisfied. This is true, provided one of the foowing two conditions is satisfied (we reca that I (i) those subdomains whose boundary intersect γ (i) (see (5)): is defined as the set of (i): card{i (i) } Ñ (Ñ being, we reca, the number of poynomias exacty reproduced in Ṽ 0 ); in this case stabiity wi hod for a 0 ; (ii): if (i) does not hod, then we need that for each n I (i) there exists ϕ,k M δ m such that supp ϕ,k Γ n; this certainy hods provided that () max inf{ : ϕ,n M (γ m ), supp ϕ,n Γ n }. (99) n I (i)
THE MORTAR METHOD IN THE WAVELET CONTEXT 671 With such a choice of the spaces, we wi consider probem (9). Since assumptions (A.1 A.3) are fufied (as we checked in the previous section), the soution u δ of such probem wi satisfy the foowing error estimate. Coroary 3.19. Let u be soution of (10), and et u δ be the soution of probem (PD-9), with the above choice of approximation and mutipier spaces. If u Ω H s (Ω ), s > 3/, then it hods u u δ X ( 1+ og h )( L ) 1/ ()(s 1) u s,ω + (100) ( + m=(,i) I =1 ()(s 1) u ν s 3/,γ m ) 1/ (101) Proof. Either one of the conditions (i) and (ii) is sufficient to ensure that assumption (BP) is satisfied with a constant C utimatey depending on the geometric decomposition of Ω in subdomains and on the particuar waveet space chosen. Since the approximation and mutipier spaces satisfy assumption (A.1 A.3), we can then appy Coroary.14, which, together with the Jackson type inequaities (91) and (74) yieds (100). Remark 3.0. In the case of a geometricay non conforming decomposition, condition (ii) might be quite restrictive, forcing the discretization to be very fine ( () N 1 inf Γ n ). It is opinion of the authors that such condition can in practice be reaxed by ony asking () inf Γ n. In the geometricay conforming case we have instead the foowing (optima) error estimate: Theorem 3.1. If the decomposition is geometricay conforming, under the assumptions of Coroary 3.19, it hods ( L ) 1/ ( u u δ X ()(s 1) u s,ω + =1 m=(,i) I ()(s 1) u ν s 3/,γ m ) 1/. (10) Proof. The resut foows from Coroary.18, provided it exits a proector ˇΠ satisfying the assumptions of Theorem.16. Indeed, observing that for a geometricay conforming decomposition a cross points are vertices of a the subdomains to which they beong, the proector ˇΠ : H s (]0, 1[ ) V can be defined as: ˇΠ (u) =P (u)+ 4 ( ) u(ai ) Ai Φ Ai (A u Φ i) Φ Ai i=1 where for each vertex A i of [0, 1] we use the notation Φ Ai to indicate the ony scaing function among the eements of the basis of V that does not vanish at A i, (that is for instance: if A 0 =(0, 0), then Φ A0 (x, y) =ϕ,0 (x)ϕ,0 (y), if A 1 =(1, 0), then Φ A1 (x, y) =ϕ,(x)ϕ,0 (y), and so on). It is easiy verified that ˇΠ is indeed a proector and that it verifies ˇΠ (u)(a i )=u(a i ) for any vertex A i. Moreover it is not difficut to check that the Jackson inequaity of the form (48) is vaid for s = R and t = N, provided ˇΠ is bounded from H t (Ω )toh s (Ω ). This is the case if 1 <t.
67 S. BERTOLUZZA AND V. PERRIER 3.5. Some remarks on the impementation It is we known that in the impementation of the mortar method, the mutipication by the stiffness matrix can be performed by appying subdomainwise the oca stiffness matrix (which does not take into account the constraints), after mutipication by a transfer matrix which gives the vaues of the constrained degrees of freedom (the ones iving on the interior of mutipier sides) in terms of the remaining (free) degrees of freedom. The transfer matrix is a discrete reaization of the proector π (Sect..5). In the case that a geometricay conforming decomposition is considered and that a subdomains are discretized by waveets, appying such proector (π, see (71)) reduces to performing either a fast waveet transform or an inverse fast waveet transform depending on which of the two discretizations on the trace and on the mutipier side is finer. In genera, when a non geometricay conforming decomposition is considered and/or waveets are couped with some other method, in the case in which a mutipier side γ m is chosen in a subdomain discretized by waveets, then, thanks to (71), appying the transfer matrix reduces to computing the scaar products of the functions on the corresponding trace sides with the scaing functions ϕ,k, k =1,...,. We refer to [16] for an anaysis of the effect that using numerica quadrature in computing the proector π has on the method. Furthermore, in the numerica resoution of the inear system arising from the waveet mortar method it is possibe to take advantage of the features of waveet basis in order to design efficient preconditioners. This can be done for instance by using the good spectra properties of waveets in an iterative substructuring approach [10]. References [1] Y. Achdou, G. Abdouaev, Y. Kutznetsov and C. Prud homme, On the parae inpementation of the mortar eement method. ESAIM: MAN 33 (1999) 45 59. [] L. Anderson, N. Ha, B. Jawerth and G. Peters, Waveets on cosed subsets on the rea ine, in Topics in the theory and appications of waveets, L.L. Schumaker and G. Webb, Eds., Academic Press, Boston (1993) 1 61. [3] F. Ben Begacem, The mortar finite eement method with Lagrange mutipier. Numer. Math. 84 (1999) 173 197. [4] F. Ben Begacem, A. Buffa and Y. Maday, The mortar eement method for 3D Maxwe s equations. C. R. Acad. Sci. Paris Sér. I Math. 39 (1999) 903 908. [5] F. Ben Begacem and Y. Maday, Non conforming spectra method for second order eiptic probems in 3D. East-West J. Numer. Math. 4 (1994) 35 51. [6] C. Bernardi, Y. Maday, C. Mavripiis and A.T. Patera, The mortar eement method appied to spectra discretizations, in Finite eement anaysis in fuids. Proc. of the seventh internationa conference on finite eement methods in fow probems, T. Chung and G. Karr, Eds., UAH Press (1989). [7] C. Bernardi, Y. Maday and A.T. Patera, Domain decomposition by the mortar eement method, in Asymptotic and numerica methods for partia differentia equations with critica parameters, H.G. Kaper and M. Garbey, Eds., N.A.T.O. ASI Ser. C 384. [8] C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: the mortar eement method, in Noninear partia differentia equations and their appications, Coège de France Seminar XI, H.Brezis and J.L.Lions, Eds. (1994) 13 51. [9] S. Bertouzza, An adaptive waveet coocation method based on interpoating waveets, in Mutiscae waveet methods for partia differentia equations. W. Dahmen, A.J. Kurdia and P. Oswad, Eds., Academic Press 6 (1997) 109 135. [10] S. Bertouzza and V. Perrier, The mortar method in the waveet context. Technica Report 99-17, LAGA, Université Paris 13 (1999). [11] S. Bertouzza and P. Pietra, Space frequency adaptive approximation for quantum hydrodynamic modes. Transport Theory Statist. Phys. 8 (000) 375 395. [1] D. Braess and W. Dahmen, Stabiity estimate of the mortar finite eement method for 3-dimensiona probems. East-West J. Numer. Math. 6 (1998) 49 64. [13] F. Brezzi and M. Fortin, Mixed and hybrid finite eement methods. Springer-Verag, New York (1991). [14] C. Canuto and A. Tabacco, Mutieve decomposition of functiona spaces. J. Fourier Ana. App. 3 (1997) 715 74. [15] C. Canuto, A. Tabacco and K. Urban, The waveet eement method. Part I: Construction and anaysis. App. Comput. Harmon. Ana. ACHA 6 (1999) 1 5. [16] L. Cazabeau, C. Lacour and Y. Maday, Numerica quadratures and mortar methods, in Computationa Sciences for the 1st Century, Bristeau et a., Eds., John Wiey & Sons, New York (1997) 119 18. [17] P. Charton and V. Perrier, A pseudo-waveet scheme for the two-dimensiona Navier-Stokes equation. Comput. App. Math. 15 (1996) 139 160.