The Wavelet Element Method Part I: Construction and Analysis Claudio Canuto y Anita Tabacco z Karsten Urban x{ This work was partially supported by th

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1 The Wavelet Element Method Part I: Construction and Analysis Claudio Canuto y Anita Tabacco z Karsten Urban x{ This work was partially supported by the following funds: in Italy, MURST ex-40% Analisi Numerica and CNR ST/74 Progetto Strategico Modelli e Metodi per la Matematica e l'ingegneria; in Germany, DAAD Vigoni{ Project Multilevel{Zerlegungsverfahren fur Partielle Dierentialgleichungen and DFG{Graduiertenkolleg Analyse und Konstruktion in der Mathematik at the RWTH Aachen. y Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy, and Istituto di Analisi Numerica del C.N.R., Pavia, Italy, e{mail: ccanuto@polito.it. z Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy, e{mail: tabacco@polito.it. x Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, Aachen, Germany, e{mail: urban@igpm.rwth-aachen.de. { This author is in particular grateful to Wolfgang Dahmen. Moreover, he thanks Gero Nieen for helpful remarks concerning presentation. 1

2 2 C. Canuto, A. Tabacco, K. Urban Proposed Running Head: THE WAVELET ELEMENT METHOD, PART I Abstract The Wavelet Element Method (WEM) combines biorthogonal wavelet systems with the philosophy of Spectral Element Methods in order to obtain a biorthogonal wavelet system on fairly general bounded domains in some IR n. The domain of interest is split into subdomains which are mapped to a simple reference domain, here n{dimensional cubes. Thus, one has to construct appropriate biorthogonal wavelets on the reference domain such that mapping them to each subdomain and matching along the interfaces leads to a wavelet system on the domain. In this paper we use adapted biorthogonal wavelet systems on the interval in such a way, that tensor products of these functions can be used for the construction of wavelet bases on the reference domain. We describe the matching procedure in any dimension n in order to impose continuity and prove that it leads to a construction of a biorthogonal wavelet system on the domain. These wavelet systems characterize Sobolev spaces measuring both piecewise and global regularity. The construction is detailed for a bivariate example and an application to the numerical solution of second order partial dierential equations is given. Keywords: Wavelet Element Method, biorthogonal wavelets on the interval, matching conditions. AMS subject classication: 42C15, 65N55, 65M70 Corresponding author: Claudio Canuto Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi, 24 I TORINO Italy phone: fax: ccanuto@polito.it

3 The Wavelet Element Method, Part I 3 1 Introduction During the past years, wavelets have become a powerful tool in both pure and applied mathematics. For example, they allow to extend classical results of Fourier Analysis to a much wider class of function spaces, [31]. On the other hand, wavelet and multilevel systems are by now very widely used in many elds of science and technology such as signal analysis, data compression and image processing, [23, 35, 33]. More recently, starting from [4], they have shown promising features for the construction of ecient numerical schemes for solving operator equations, see e.g. [17]. Many constructions of wavelets can be found in the literature. Each of them provide dierent features such as smoothness, arbitrary degree of exactness of approximation, compact support in physical or transformed space, etc. However, currently, most of these constructions are restricted to \simple domains", namely IR n, the torus, the n{dimensional cube, or domains that can be easily mapped to these ones. This is a severe limitation to the successful use of wavelets in certain elds. Only in the last few years papers have appeared aimed at dealing with wavelets in general bounded domains, [25, 10, 22]. In this paper, we propose a construction of biorthogonal wavelet systems on fairly general bounded domains, by following the philosophy that led A.T. Patera [32] to invent the Spectral Element Method (SEM). The SEM uses a global, high order polynomial basis on a closed interval, and, by tensor product, extends it on a n{dimensional cube. The construction on a bounded domain having complex geometry is then carried on by splitting the domain into subdomains and mapping these to a single reference domain, namely a cube. This has led to very ecient numerical solvers for partial dierential equations, with signicant applications also for \real life problems", [29]. The key for the eciency of the SEM is the tensor product structure of the basis on the reference domain. Since nowadays mathematically sound and computationally ecient univariate wavelet systems are available on a closed interval, we propose to replace the global polynomial basis by such a multiscale basis; then we apply the above splitting-and-mapping approach, adding the advantage of multiscale decompositions to those of the SEM. The resulting construction provides a multilevel decomposition for function spaces built on the domain; so, it can be applied to any circumstance in which this is needed. In particular, as for the SEM, the numerical approximation of operator equations can be a challenging eld of application. The motivation for using wavelets here is at least twofold: they provide optimal preconditioning of the arising ill conditioned linear systems [24, 18, 21] and they allow the denition of ecient adaptive schemes, [28, 30, 3, 13, 5, 14]. In addition, the exibility in the construction of biorthogonal wavelets leaves some room which can be used to adapt these systems to special problems at hand, see [15, 34, 19] for example. Biorthogonal wavelet systems on the unitary interval, which can be required to satisfy certain boundary conditions, are the initial point of our construction. The univariate systems are de- ned for instance as in [20, 27], starting from systems on the real line such as, e.g., Daubechies' compactly supported orthogonal wavelets [23] or the biorthogonal B{spline wavelets [11]. We recall that dealing with multiscale methods involves two dierent bases for the trial spaces, namely the single scale and the multiscale (or detail-) basis. The single scale basis is similar to nite elements on uniformly rened triangles or global polynomial bases on cubes. Hence, the matching of the single scale basis functions along the interfaces of the subdomains is similar to the matching in the SEM. The multiscale basis can be understood to span the details between succeeding trial spaces. For these functions (named wavelets) matching is more delicate. More-

4 4 C. Canuto, A. Tabacco, K. Urban over, preconditioning and adaptivity is based on certain stability properties of the wavelet bases which have to be valid also after the matching. In this paper, we aim at designing bases that can be used, e.g., to build trial spaces in a Galerkin projection method for approximating second order partial dierential equations; hence, we enforce a conformal C 0 {matching. We prove that we can match wavelet functions in an appropriate way and give the construction independently of the spatial dimension. Other kind of matchings with dierent level of non{conformity will be considered elsewhere. A preliminary application and one particular example is given. In a forthcoming paper, [7], we shall address many issues related to the actual realization in dimension 2 and 3 and provide other applications and properties. The paper is organized as follows: In x2, we review the main properties of biorthogonal wavelet systems on the interval and describe the possibilities to add certain boundary conditions to these systems. This latter topic is discussed in the Appendix A in more detail for the convenience of the reader. By using tensor products one can then easily obtain wavelet systems on n{ dimensional cubes. Section 3 is devoted to the description of this construction. Moreover, we recall that stability can easily be carried over from the corresponding univariate property. In x4, we use these multiscale bases on cubes to obtain a multiresolution decomposition on a general bounded n{dimensional domain partitioned into subdomains. The construction of biorthogonal wavelets is introduced in x5. The method is detailed for a bivariate example in x5.3 and the main results are collected in x5.4. Finally, x6 contains an application to the numerical solution of elliptic second order partial dierential equations. Some of the results of this paper (in x4 and Appendix A) overlap with similar ones by Dahmen and Schneider, [22]. Although we stemmed the idea of the Wavelet Element Method (WEM) from the SEM independently of these authors, we are indebted to Wolfgang Dahmen for many discussions on multiscale methods over the past years, that have been quite inuencial for us. On the other hand, our construction of the WEM, in particular of the matching of the wavelet functions, diers from the results in [22]. 2 Biorthogonal systems on [0; 1] In the literature, there is a whole variety of concrete examples of multiresolution analyses on the interval. All these constructions are based on scaling functions on the real line that are either orthogonal or biorthogonal. Then, these functions are modied near the boundaries in order to ensure the validity of this and other conditions on the interval, see [2, 12, 8, 20, 27] for example. In this section we collect the main properties of the biorthogonal wavelet systems on the interval, as constructed in [20] or in [27]. All the results we give are proven in these references, except a small number of them whose proofs will be provided in the Appendix. We rst describe the general approach and then we detail the modications for fullling boundary conditions. We will frequently use the following notation: by A < B we denote the fact that A can be bounded by a multiple constant times B, where the constant is independent of the various parameters A and B may depend on. Furthermore, A < B < A (with dierent constants, of course) will be abbreviated by A B. 2.1 General setting The starting point are two families of scaling functions j := f j;k : k 2 j g; ~ j := f ~ j;k : k 2 j g L 2 (0; 1);

5 The Wavelet Element Method, Part I 5 where j denotes an appropriate set of indices and j j 0 can be understood as the scale parameter (with some j 0 denoting the coarsest scale). For subsequent convenience, these functions will not be labeled by integers as usual, but rather by a set of real indices j := f j;1 ; : : : ; j;kj g; 0 = j;1 < j;2 < < j;kj = 1: (2.1) In other words, each basis function is associated with a node, or grid point, in the interval [0; 1]; the actual position of the internal nodes j;2 ; : : : ; j;kj?1 will be irrelevant in the sequel, except that it is required that j j+1 (see (2.3-k)). It will be also convenient to consider j as the column vector ( j;k ) k2 j, and analogously for other set of functions. The construction of these families j, ~ j guarantees that they are dual generator systems of a multiresolution analysis in L 2 (0; 1) S j := span j ; ~ Sj := span ~ j ; (2.2) in the sense that the following conditions in (2:3) are fullled: (2.3-a) The systems j and ~ j are renable, i.e., there exist matrices M j ; ~ Mj, such that j = M j j+1 ; ~ j = ~ Mj ~ j+1 : This implies, in particular, that the induced spaces S j, ~ Sj are nested, i.e., S j S j+1, ~ Sj ~ Sj+1. (2.3-b) The functions have local support, in the sense (2.3-c) The systems are biorthogonal, i.e., (2.3-d) The functions are regular, i.e., diam (supp j;k ) diam (supp ~ j;k ) 2?j : ( j;k ; ~ j;k 0) L 2 (0;1) = k;k 0; for all k; k 0 2 j : j;k 2 H (0; 1); ~ j;k 2 H ~ (0; 1); for some ; ~ > 1; where H s (0; 1), s 0, denotes the usual Sobolev space on the interval as dened, e.g., in [1]. (2.3-e) The systems are exact of order L, ~ L 1, respectively, i.e., polynomials up to the degree L? 1, ~ L? 1 are reproduced exactly: IP L?1 (0; 1) S j ; IP ~ L?1 (0; 1) ~ Sj ; where IP r (0; 1) denotes the set of the algebraic polynomials of degree r at most, restricted to [0; 1]. (2.3-f) The systems j, ~ j are uniformly stable, i.e., where c := (c k ) k2 j. k2j c k j;k L 2 (0;1) kck`2(j ) k2j c k ~ j;k L 2 (0;1);

6 6 C. Canuto, A. Tabacco, K. Urban (2.3-g) The operators j : L 2 (0; 1)! S j dened by j v := (v; ~ j;k ) L 2 (0;1) j;k k2j have the properties j j+1 = j, 2 j = j, k j k < 1, and analogously for ~ j. with (2.3-h) The systems j, ~ j fulll a Jackson{type inequality: inf v j 2S j kv? v j k L 2 (0;1) < 2?sj kvk H s (0;1) ; v 2 H s (0; 1); 0 s min(l; ); inf v j 2 ~ S j kv? v j k L 2 (0;1) < 2?sj kvk H s (0;1) ; v 2 H s (0; 1); 0 s min(~ L; ~): (2.3-i) The systems j, ~ j fulll a Bernstein{type inequality: kv j k H s (0;1) < 2 js kv j k L 2 (0;1); v j 2 S j ; 0 s ; kv j k H s (0;1) < 2 js kv j k L 2 (0;1); v j 2 ~ Sj ; 0 s ~: (2.3-j) There exist complement spaces T j and ~ Tj such that S j+1 = S j T j ; ~ Sj+1 = ~ Sj ~ Tj ; T j? ~ Sj ; ~ Tj? S j : (2.3-k) The spaces T j and ~ Tj have biorthogonal, stable bases (in the sense of (2.3-f)) j = f j;h : h 2 r j g; ~ j = f~ j;h : h 2 r j g; r j := j+1 n j = f j;1 ; : : : ; j;mj g; 0 < j;1 < < j;mj < 1: These basis functions are called biorthogonal wavelets. (2.3-l) The collections of these functions for all j j 0 form Riesz bases of L 2 (0; 1). Even more than that, these systems admit norm equivalences for a whole range in the Sobolev scale: k2j 0 c j0 ;k j0 ;k + 1 d j;h j;h 2 h2r j s j=j 0 k2j 0 jc j0 ;kj j=j 0 h2r j 2 2sj jd j;h j 2 ; where s 2 (? min(~ L; ~); min(l; )) is related to the regularity and the exactness of the generator system, and s = H s (0; 1) if s 0 or s = (H?s (0; 1)) 0 if s < 0. The following concept will be important in the sequel. The system j is said to be reection invariant, if j is invariant under the mapping x 7! 1? x and which can also be abbreviated as j;k (1? x) = j;1?k (x); for all x 2 [0; 1] and k 2 j ; (2.4) j (1? x) = l j (x): A similar denition can be given for the system j, as well as for the dual systems. If j is reection invariant, then j can be built to have the same property. This will be always implicitly assumed. For example, reection invariant systems can be constructed from biorthogonal B{ splines [11], whereas this is not possible starting from compactly supported Daubechies' scaling functions [23], as they lack of symmetry.

7 The Wavelet Element Method, Part I Systems fullling boundary conditions One may want to incorporate boundary conditions in a multiresolution analysis, which will be crucial for the further construction of the WEM. To this end, let us introduce the following denitions. Denition 2.1 The systems j and j are called boundary adapted if, at each boundary point: (i) only one basis function in each system is not vanishing; precisely, j;k (0) 6= 0 () k = 0; j;k (1) 6= 0 () k = 1; (2.5) j;h (0) 6= 0 () h = j;1 ; j;h (1) 6= 0 () h = j;mj ; (2.6) (ii) the nonvanishing scaling and wavelet functions take the same value; precisely, there exist constants c 0 and c 1 independent of j such that j;0 (0) = j;j;1 (0) = c 0 2 j=2 ; j;1 (1) = j;j;mj (1) = c 1 2 j=2 : (2.7) Denition 2.2 The system j is called boundary symmetric if j;0 (0) = j;1 (1): (2.8) Note that if the systems j and j are boundary adapted and if j is boundary symmetric, then also j has the same property. From now on, we shall assume that the systems j, j and ~ j, ~ j are both boundary adapted and boundary symmetric. As far as the former assumption is concerned, starting from generator and wavelet systems on the interval, one can indeed construct boundary adapted ones, for instance, for orthogonal systems and for systems arising from biorthogonal B{splines (see Proposition A.5 and Proposition A.6 in the Appendix, and also [22]). On the other hand, the latter assumption is not strictly necessary for carrying on our construction, yet it will greatly simplify the subsequent formalism. It holds for all reection invariant systems, such as biorthogonal B{splines. Boundary adapted generator and wavelet systems can be modied in order to fulll homogeneous Dirichlet boundary conditions. For the scaling functions this is easily done by omitting those functions that do not vanish at those end points of the interval where boundary conditions are enforced. For the wavelets, the situation is a little bit more involved. To be specic, let us rst introduce the following sets of the internal grid points: int j := j n f0; 1g; r int j := r j n f j;1 ; j;mj g: (2.9) Let us collect in the vector = ( 0 ; 1 ) 2 f0; 1g 2 the information about where homogeneous boundary conditions are enforced, i.e., d = 1 means no boundary condition, whereas d = 0 denotes boundary condition at the point d 2 f0; 1g. The corresponding set of indices is then given by j := 8 > < >: int j ; if = (0; 0); j n f0g; if = (0; 1); j n f1g; if = (1; 0); j ; if = (1; 1): (2.10)

8 8 C. Canuto, A. Tabacco, K. Urban Let the generator systems be dened as and let us dene the multiresolution analyses j := f j;k : k 2 j g; ~ j := f~ j;k : k 2 j g; S j := span j ; ~ S j := span ~ j : (2.11) The associated biorthogonal wavelet systems j, ~ j are the same as the boundary adapted ones except that we possibly change the rst and/or the last wavelet depending on. More precisely, the wavelets can be chosen to vanish at each boundary point in which the corresponding component of is zero. If the boundary condition is prescribed at 0, the rst wavelets j;j;1 and ~ j;j;1 are replaced by D j; j;1 := 1 p 2 ( j;j;1? j;0 ); ~ D j; j;1 := 1 p 2 (~ j;j;1? ~ j;0 ); (2.12) respectively. The wavelets D j; j;mj and ~ D j; j;mj vanishing at 1 are dened similarly. We refer to Appendix A (see Corollary A.7) for the detailed construction of the new wavelet systems. Observe that the set of grid points r j which labels the wavelets does not change, i.e., r j = r j for all choices of. The new systems j, j and ~ j, ~ j fulll the conditions in (2:3) stated above, provided the index is appended to all symbols. To be more precise, in (2.3-e) the space of polynomials IP r (0; 1) is dened as IP r (0; 1) := fp 2 IP r (0; 1) : p(d) = 0 if d = 0; for d = 0; 1g; in (2.3-g), the projection operators j are dened as j v := (v; ~ j;k ) L 2 (0;1) j;k ; k2 j nally, in (2.3-d) the Sobolev spaces H s (0; 1) are dened as H s (0; 1) := fv 2 H s (0; 1) : v(d) = 0 if d = 0; for d = 0; 1g; (2.13) for s 2 IN n f0g, and by interpolation for s 62 IN; s > 0. Note that, unlike a common notation for Sobolev spaces with boundary conditions, we only require the vanishing of v, not of its derivatives, even in the case s > 3=2. Finally, suppose that the systems j and ~ j are reection invariant, see (2:4). Then, the systems with boundary conditions can be built to be reection invariant as well, in an obvious sense (i.e., the mapping x 7! 1? x induces a mapping of j into itself if = (0; 0) or = (1; 1), while it produces an exchange of (0;1) j with (1;0) j in the other cases). 3 Tensor products The perhaps simplest way to build multivariate wavelets based on univariate ones is to employ tensor products. In this section we set up the notation for biorthogonal multiresolution analyses

9 The Wavelet Element Method, Part I 9 in ^, where ^ = (0; 1) n, and we collect some properties that are well known. The notation in this section is already taylored to the kind of application of this material in the rest of the paper, namely, using ^ as a reference domain. Let us x a vector b = ( 1 ; : : : ; n ) containing the information on the particular boundary conditions, where each l 2 f0; 1g 2 for 1 l n. Let us set, for all j j 0, Vj b (^) := S 1 j S n j ; and similarly for V ~ b j (^). In order to construct a basis for these spaces, we dene for ^x = (^x 1 ; : : : ; ^x n ) 2 ^ and ^k = (^k 1 ; : : : ; ^k n ) 2 b j := 1 j n j we set so that ^' j;^k (^x) := ( j;^k 1 j;^k n )(^x) = V b j (^) = span (^ j ); ^ j := f ^' j;^k : ^k 2 b jg; ny l=1 j;^k l (^x l ); ~ V b j (^) = span (^~ j ): Similarly to the univariate case, let us introduce the function spaces H s b (^) by H s b (^) := f^v 2 H s (^) : ^v jf^xl =dg 0 if l d = 0; (3.1) for l = 1; : : : ; n; d = 0; 1g; for s 2 IN n f0g, and by interpolation for s 62 IN; s > 0. Then, we note that V b j (^) H b (^); ~ V b j (^) H ~ b (^): It is trivially seen that these spaces are nested, i.e., Vj b(^) V b j+1 (^), V ~ b j (^) V ~ b j+1 (^). For the corresponding projectors, we dene ^P b j ^v := ( 1 j;1 n j;n )^v = ^k2 b j (^v; ^~' j;^k ) L 2 ( ^) ^' j;^k ; (3.2) where l j;l denotes the application of j with respect to the direction l, 1 l n and = l. Indeed, using the induction principle, (3:2) can be seen by the following reasoning ( 1 j;1 2 j;2 )^v = 2 j;2 1 j;1^v(^x 1; ) = 2 j;2 = ^k j ^k j ^k j ^v(^x 1 ; ); ~ j;^k2 L 2(0;1) j;^k2 (^v; ~ j;^k 1 ~ j;^k 2 ) L 2 ((0;1) 2 ) j;^k 1 j;^k 2 : Moreover, the properties k ^P b j k < 1, ^P b j ^P b j+1 = ^P b j and ( ^P b j ) 2 = ^P b j easily follow from the tensor product structure of ^P b j. The polynomial exactness is trivial and the stability is implied by the biorthogonality ( ^' j;^k ; ^~' j;^k 0 ) L 2 ( ^) = ^k;^k 0 ; ^k; ^k0 2 b j;

10 10 C. Canuto, A. Tabacco, K. Urban and the locality of the generators, [16, 20]; it can also directly be checked by the following reasoning (here, for simplicity we set n = 2) ^k2 b j c^k ^' j;^k L 2 ((0;1) 2 ) = ^k j ^k j ^k j kfc^k g^k2 b j k`2( b j ): c^k j;^k 1 j;^k 2 L 2 (0;1;L 2 (0;1)) c^k j;^k 1 ^k j L 2 (0;1;`2( 2 j )) The Jackson and Bernstein inequalities, which extend in an obvious way (2.3-h) and (2.3-i), are well known to be implied by general principles [16, 20], but they can also be directly deduced by the univariate properties (here for simplicity for n = 2): for each ^v 2 H s b ((0; 1)2 ) with 0 s min(l; ) b k^v? ^P j ^vk L 2 ((0;1) 2 ) k^v? 1 j;1^vk L 2 ((0;1) 2 ) + k 1 (^v? 2 < j;1 j;2^v)k L 2 ((0;1) 2 ) 2?sj k^vk H s (0;1;L 2 (0;1)) + k^v? 2 j;2^vk L 2 ((0;1) 2 ) < 2?sj k^vk H s ((0;1) 2 ): Let us now consider complement spaces W b j (^) and ~ W b j (^) such that V b j+1(^) = V b j (^) W b j (^); W b j (^)? ~ V b j (^); V ~ b j+1 (^) = V ~ b j (^) W ~ b j (^); W ~ b j (^)? Vj b (^): Let us set r b j := b j+1 n b j. Given any ^h = (^h 1 ; : : : ; ^h n ) 2 r b j, we dene the corresponding wavelet ny ^j;^h (^x) := ( ^#^h 1 ^#^h n )(^x) = ^#^h l (^x l ); where ^#^h l := 8 < : j;^h l ; if ^h l 2 l j;^h l ; if ^h l 2 r j, j, l=1 and we set ^ j := f ^j;^h : ^h 2 r b jg; W b j (^) := span ^ j: A parallel construction is done for the dual complement space ~ W b j (^). Due to the univariate properties, we have ( ^' j;^k ; ^~ j;^h ) L 2 ( ^) = 0; ( ^j;^h ; ^~' j;^k ) L 2 ( ^) = 0; 8^k 2 b j ; 8^h 2 r b j ; as well as ( ^j;^h ; ^~ j 0 ;^h 0 ) L 2 ( ^) = jj 0^h ^h 0 ; 8j; j 0 j 0 ; 8^h 2 r b j; 8^h 0 2 r b j 0: Moreover, the wavelets form a Riesz basis in L 2 (^) and the norm equivalences (2.3-l) extend to the multivariate case.

11 The Wavelet Element Method, Part I 11 Finally, considering the boundary values, we note that, given any l 2 f1; : : : ; ng and d 2 f0; 1g, we have ( ^' j;^k ) j^x l =d 0 i ^kl 6= d or (^k l = d and l d = 0) and ( ^j;^h ) j^x l =d 0 i ^hl 6= d or (^h l = d and l d = 0); with d = j;1 if d = 0, and d = j;mj if d = 1. 4 Multiresolution on general domains Recently, various constructions of multilevel decompositions on general bounded domains have been introduced, [10, 22]. Some aspects of the latter one are closely related to our approach. The idea is to subdivide the domain of interest IR n into subdomains i, which are images of the reference element ^ = (0; 1) n. The multiresolution analysis on is then obtained by transformations of properly matched systems on ^. Let us rst set some notation, starting with the reference domain ^. For 0 p n? 1, a p{ face of ^ is a subset ^ dened by the choice of a set L^ of indices l 1 ; : : : ; l n?p 2 f1; : : : ; ng and a set of integers d 1 ; : : : ; d n?p 2 f0; 1g in the following way ^ = f(^x 1 ; : : : ; ^x n ) : ^x l1 = d 1 ; : : : ; ^x ln?p = d n?p ; and 0 ^x l 1 if l 62 L^ g (4.1) (thus, e.g., in 3D, a 0{face is a vertex, a 1{face is a side and a 2{face is a usual face of the reference cube). The coordinates ^x l with l 2 L^ will be termed the frozen coordinates of ^, whereas the remaining coordinates will be termed the free coordinates of ^. Let ^ and ^ 0 be two p{faces of ^, and let H : ^! ^ 0 be a bijective mapping. We shall say that H is order{preserving if it is a composition of elementary permutations (s; t) 7! (t; s) of the free coordinates of ^. An order{preserving mapping is a particular case of an ane mapping, as made precise by the following simple Lemma. Lemma 4.1 H is ane if and only if it is a composition of elementary permutations (s; t) 7! (t; s) and reections s 7! 1? s of the free coordinates of ^. Proof. If H is order{preserving it is trivially seen that H is ane. Conversely, by neglecting the frozen coordinates, we can assume that H : [0; 1] p! [0; 1] p ; then, we use induction on p. If p = 1, the result is obvious. Otherwise, set H 0 := H([0; 1] p?1 f0g) and H 1 := H([0; 1] p?1 f1g). If H is ane, then H([0; 1] p ) = f(1? s)h 0 + sh 1 : s 2 [0; 1]g = [0; 1] p, since H is bijective. Thus, necessarily, H 0 is a (p? 1){face, say H 0 = f^x 0 l = dg for some l 2 f1; : : : ; pg and some d 2 f0; 1g, whereas H 1 = f^x 0 l = 1? dg. It follows that H maps ^x p into ^x 0 l = ^x p (if d = 0) or ^x 0 l = 1? ^x p (if d = 1). Since H : [0; 1] p?1 f0g! H 0 is ane and bijective, we conclude by induction. Let us now consider our domain of interest IR n, with Lipschitz We assume that there exist N open disjoint subdomains i (i = 1; : : : ; N) such that = N[ i=1 i

12 12 C. Canuto, A. Tabacco, K. Urban and such that, for some r (see (2.3-d)), there exist r{time continuously dierentiable mappings F i : ^! i (i = 1; : : : ; N) satisfying i = F i (^); det(jf i ) > 0 in ^; where JF i denotes the Jacobian of F i ; in the sequel, it will be useful to set G i := F i?1. The image of a p{face of ^ under the mapping F i will be termed a p{face of i ; if? i;i 0 i i 0 is nonempty for some i 6= i 0, then we assume that? i;i 0 is a p{face of both i and i 0 for some 0 p n? 1. In addition, setting? i;i 0 = F i (^) = F i 0(^ 0 ); with two p{faces ^ and ^ 0 of ^, we require that the bijection fullls the following Hypothesis (4.2): a) H i;i 0 is ane; H i;i 0 := G i 0 F i : ^! ^ 0 b) in addition, if the systems of scaling functions and wavelets on [0; 1] are not reection invariant (see (2:4)), then H i;i 0 is order{preserving. Remark 4.2 Suppose that Hypothesis (4:2) holds true. It is easily seen that, if n = 2, it is always possible to modify the mappings F i in such a way that the new mappings H i;i 0 are all order{preserving. However, this is not true if n = 3, as the example of a 3D Moebius ring indicates. The is subdivided in two relatively open parts (with respect the Dirichlet part? D and the Neumann part? N, in such a way =? D [? N ;? D \? N = ;; where for i = 1; : : : ; N we suppose i \? D i \? N are (possibly empty) unions of p{faces of i. 4.1 Multiresolution and wavelets on the subdomains Let us now introduce multiresolution analyses on each i, i = 1; : : : ; N, by \mapping" appropriate multiresolution analyses on ^. To this end, let us dene the vector b( i ) = ( 1 ; : : : ; n ) 2 f0; 1g 2n as follows l d = ( 0; if Fi (f^x l = dg)? D ; 1; otherwise, l = 1; : : : ; n; d = 0; 1: Moreover, let us introduce the one-to-one transformation v 7! ^v := v F i ;

13 The Wavelet Element Method, Part I 13 which maps functions dened in into functions dened in ^. Next, for all j j 0, let us set V j ( i ) := fv : ^v 2 V b( i) j (^)g: If we introduce, for any s 0, the Sobolev spaces (see (3:1)), we observe that The projection operators P i j H s b ( i ) = fv : ^v 2 H s b(i) (^)g V j ( i ) H b ( i): (4.3) : L 2 ( i )! V j ( i ) are dened by the commutativity relation d (P i b(i) j v) := ^P j ^v; 8v 2 L 2 ( i ): This denition suggests to equip L 2 ( i ) by the inner product Z Z hu; vi := i u(x) v(x) jjg i (x)j dx = ^u(^x) ^v(^x) d^x; (4.4) i which, due to the properties of the transformation of the domains, induces an equivalent L 2 {type norm kvk 2 L 2 (i) hv; vi i = k^vk 2 L 2 ( ^) ; 8v 2 L2 ( i ): Let us now dene the single scale basis functions for the above dened multiresolution spaces. To this end, for i = 1; : : : ; N, let us consider the set b( i) j of grid points in ^, and let us dene and k (i) := F i (^k); K i j := fk (i) ^ ^k 2 b(i) j : ^k 2 b(i) j g: In this way, we have a set of grid points in i. Each grid point can be associated to a basis function in V j ( i ). Precisely, for each k 2 K i j let us set ^k = ^k(i) := G i (k) and let us dene the function ' (i) j;k := ^' j;^k G i; i.e., d ' (i) j;k = ^' j;^k. The set of these functions will be denoted by i j. This set and the dual set ~ i j form biorthogonal bases of V j ( i ) and ~ Vj ( i ), respectively, with respect to the inner product (4:4); indeed h' (i) j;k ; ~'(i) j;k 0 i i = ( ^' j;^k ; ^~' j;^k 0 ) L 2 ( ^) = k;k 0; ^k = Gi (k); ^k0 = G i (k 0 ): (4.5) This yields the following representation of P i j : P i j v = k2k i j hv; ~' (i) j;k i i '(i) j;k = ^k2 b( i ) j (^v; ^~' j;^k ) L 2 ( ^) ^' j;^k G i: It is easily seen that the dual multiresolution analyses on i dened in this way inherit the properties of the multiresolution analyses on ^ as far as stability of bases, properties of the

14 14 C. Canuto, A. Tabacco, K. Urban biorthogonal projectors, Jackson and Bernstein inequalities (and consequent characterization of function spaces) are concerned. Obviously, the property of exact reconstruction of polynomials has to be replaced by the property of exact reconstruction of the images of polynomials under the transformation F i. For subsequent reference, we report the Jackson and Bernstein inequalities in i : kv? P i j vk L 2 (i) < 2?sj kvk H s ( i); 8v 2 H s b ( i ); 0 s min(l; ); (4.6) kvk H s ( i) < 2 sj kvk L 2 (i); 8v 2 V j ( i ); 0 s : (4.7) Finally, we come to the detail spaces. The biorthogonal complement of V j ( i ) in V j+1 ( i ) is W j ( i ) := fw : ^w 2 W b( i) j (^)g: A biorthogonal basis in this space is associated to the grid through the relation H i j := K i j+1 n K i j = fh = F i (^h) : ^h 2 r b( i) j g (4.8) (i) j;h := ^j;^h G i; 8h 2 H i j : (4.9) The set of such functions will be denoted by i j, and the dual set by ~ i j. 4.2 Multiresolution on the global domain Now we describe the construction of dual multiresolution analyses on. Let us dene, for all j j 0, V j () := fv 2 C 0 () : v j i 2 V j( i ); i = 1; : : : ; Ng; (4.10) the dual spaces Vj ~ () are dened in a similar manner, simply by replacing each V j ( i ) by Vj ~ ( i ). Then, nestedness is obvious from the analogous property in each i. We shall now dene an appropriate functional setting for the above family of spaces. To this end, let us introduce the Sobolev spaces Hb s () by H s b () := fv 2 H s () : v = 0 on? D g (4.11) for s 2 IN n f0g, and by interpolation for s 62 IN; s > 0. Furthermore, we introduce another scale of Sobolev spaces, depending upon the partition P := f i : i = 1; : : : ; Ng of ; precisely, we set H s b (; P) := fv 2 H 1 b () : v j i 2 Hs ( i ); i = 1; : : : ; Ng (4.12) for any integer s 2 INnf0g, and we extend the denition using interpolation for any s 62 IN; s > 0. Note that, for any real s 1, Hb s (; P) can indeed be dened directly by (4:12); moreover, kvk H s b (;P) N i=1 kv j i k H s (i); 8v 2 H s b (; P): In addition, Hb s() Hs b (; P) for all s 0, and Hs b () = Hs b (; P) for all s satisfying 0 s < 3=2. Recalling (4:3), one has V j () H b (; P): (4.13)

15 The Wavelet Element Method, Part I 15 In order to dene a basis of V j (), let us introduce the set K j := N[ i=1 K i j (4.14) containing all the grid points in. The following remark will be useful in the sequel. Remark 4.3 Suppose that? i;i 0 i i 0 6= ; is a p{face, i.e.,? i;i 0 = F i (^) = F i 0(^ 0 ) for two p{faces ^ and ^ 0 of ^. If k 2? i;i 0 \ K j, there exist ^k (i) 2 ^ \ b( i) j and ^k (i0 ) 2 ^ 0 \ b( i 0 ) j such that k = F i (^k (i) ) = F i 0(^k (i0 ) ) and the free coordinates of ^k (i0 ) are a permutation and (possibly) a reection of the free coordinates of ^k (i). This is a straightforward consequence of Hypothesis (4:2). Each grid point of K j can be associated to one single scale basis function of V j (), and conversely. To accomplish this, let us set o I(k) := ni 2 f1; : : : ; Ng : k 2 i ; 8k 2 K j ; as well as ^k (i) := G i (k); 8i 2 I(k); 8k 2 K j : Then, for any k 2 K j let us dene the function ' j;k as follows if i 2 I(k); ( ' ji(k)j j;k :=?1=2 ' (i) j;k ; j i 0; otherwise. (4.15) This function belongs to V j (), since it is continuous across the interelement boundaries. This is a consequence of assumptions (2:5), (2:8) and Remark 4.3. Indeed, if k 2 i (remember that i is open) for some i, then I(k) = fig and ' j;k vanishes i, therefore it is continuous. Suppose now that k belongs to a common face of subdomains, i.e., as before, k 2? i;i 0 i i 0 = F i (^) = F i 0(^ 0 ) for two p{faces ^ and ^ 0 of ^. Let x be any point of? i;i 0, and let ^x 2 ^ and ^x 0 2 ^ 0 be such that x = F i (^x) = F i 0(^x 0 ). Then, and ' j;k (x) = j i ji(k)j?1=2 j;^k (^x (i) 1 ) (^x 1 j;^k n) (4.16) n (i) ' j;k ji 0 (x) = ji(k)j?1=2 j;^k (i0 )(^x 0 1) 1 j;^k (i0 ) (^x0 n): (4.17) n Now, in (4:16) there are exactly n?p factors of type j;^k (^x (i) l ) corresponding to frozen coordinates l in ^; similarly, in (4:17), there are exactly n?p factors of type j;^k (i0 ) l (^x 0 l ) corresponding to frozen coordinates in ^ 0. By assumption (2:8), all these factors are equal. The remaining p factors in (4:16) appear in (4:17) as well, possibly in a dierent order, due to Hypothesis (4:2). Thus, ' j;k j i (x) = ' j;k j i 0 (x) for all x 2? i;i 0. Finally note that ' j;k is identically zero on each face of subdomains which does not contain k. Let us set j := f' j;k : k 2 K j g. The dual family ~ j := f ~' j;k : k 2 K j g is dened as in (4:15), simply by replacing each ' (i) j;k by ~'(i) j;k. Then, we have V j () = span j ; ~ Vj () = span ~ j :

16 16 C. Canuto, A. Tabacco, K. Urban By dening the L 2 {type inner product on hu; vi := N i=1 it is easy to obtain the biorthogonality relations h' j;k ; ~' j;k 0i = (ji(k)j ji(k 0 )j)?1=2 hu; vi i ; (4.18) h' (i) j;k ; ~'(i) i2i(k)\i(k 0 ) j;k 0 i i = k;k 0; (4.19) from the analogous relations (4:5) in each i ; indeed, I(k) \ I(k 0 ) 6= ; if and only if there exists an index i 2 f1; : : : ; Ng such that k; k 0 2 i. It is easy to check that for each k 2 K j, diam supp ' j;k 2?j, k' j;k k L < 2 () 1 and cardfk 0 : supp ' j;k \ supp ' j;k 0 6= ;g < 1. Similar results hold for the dual system ~ j. Thus, thanks to abstract results about the stability of biorthogonal bases (see, for example, [20]), we have j (~ j ; resp:) is a stable basis in V j () ( ~ Vj (); resp:): Let us introduce the biorthogonal projection operator upon V j () Pj v := hv; ~' j;k i ' j;k ; 8v 2 L 2 (): k2k j The properties P j v = v; P j P j+1 v = P j v; 8v 2 V j (); 8v 2 L2 (); kp j k L(L 2 ();L 2 ()) < 1; (and the dual ones) are obvious by the construction of the spaces V j () and their basis j. It is useful to compare Pj v with P i j v in i (i = 1; : : : ; N). We have Pj v j i = ji(k)j?1=2 hv; ~' j;k i ' (i) j;k k2k i j = ji(k)j?1 hv; ~' (i0 ) ' (i) j;k : (4.20) Thus, setting R (i) j where k2k i j i 0 2I(k) j;k i i 0 v := P i j v? Pj v and recalling that ji(k)j = 1 if k 2 i, we obtain ji r (i) j;k := ji(k)j?1 R (i) j v = i 0 2I(k) k2k j \@i [hv; ~' (i) r (i) j;k '(i) j;k ; (4.21) j;k i i? hv; ~' (i0 ) j;k i i 0 ]: We shall now establish a Jackson{type inequality for Pj. To this end, we need the three following lemmata.

17 The Wavelet Element Method, Part I 17 Lemma 4.4 Suppose that, for some l; m 2 f1; : : : ; Ng,? l;m l m is an (n? 1){face. Then, for any v 2 Hb 1() such that v 2 jl Hs ( l ) and v j m 2 H s ( m ), with 1 s min(l; ), we have jhv; ~' (l) i j;k l? hv; ~' (m) i j;k mj 2 < 2?2sj [kvk 2 H s + (l) kvk2 H s (m) ]: k2k j \?l;m Proof. By our Hypothesis (4:2), it is not restrictive to assume that? l;m = F l (^) = F m (^ 0 ), where ^ = f(0; ^x 0 ) : ^x 0 2 [0; 1] n?1 g and ^ 0 = f(1; ^x 0 ) : ^x 0 2 [0; 1] n?1 g. In addition, there is a set ^K j [0; 1] n?1 such that for all k 2 K j \? l;m, k = F l ((0; ^k )) = F m ((1; ^k )) for some ^k 2 ^K j. Consequently, ' j;kjl = ^' j;(0;^k ) G l with ^' j;(0;^k ) (^x) = j;0(^x 1 ) ^' j;^k (^x 0 ), where ^' j;^k is a tensor product of (n? 1){univariate scaling functions. Analogously, ' j;k jm = ^' j;(1;^k ) G m with ^' j;(1;^k ) (^x) = j;1(^x 1 ) ^' j;^k (^x 0 ). Similar representations hold for the dual functions ~' j;k. Let us set ^v (l) = v G j l l, ^v (m) = v j m G m. With these notations jhv; ~' (l) i j;k l? hv; ~' (m) i j;k mj 2 (4.22) k2k j \?l;m = j(^v (l) ; ^~' )? j;(0;^k ) L 2 ( ^) (^v(m) ; ^~' ) j;(1;^k ) L 2 ( ^) j2 : ^k 2 ^K j Now, we apply the inequality kfk L 2 (@ kfk1=2 ^) L 2 kfk1=2, which holds for all f 2 ( ^) H 1 ( ^) H1 (^), to f = ^v (l)? ^P b j ^v (l), with b = b( l ). Thanks to the characterization of H 1 (^) associated to the wavelet system in ^, we have k^v (l)? ^P b j ^v (l) k H 1 ( ^) < 2?(s?1)j k^v (l) k H s ( ^) ; together with the Jackson inequality in L 2 (^), we obtain k^v (l)? ^P b j ^v (l) k L 2 (^) 2?(s?1=2)j k^v (l) k H s ( ^) : (4.23) Similarly, k^v (m)? ^P b j ^v (m) k L 2 (^) 2?(s?1=2)j k^v (m) k H s ( ^) : (4.24) Note that the functions ^x 0 7! ^v (l) ((0; ^x 0 )) and ^x 0 7! ^v (m) ((1; ^x 0 )) coincide (in the sense of L 2 ((0; 1) n?1 )), since v and v j l jm have a common trace on? l;m. Moreover, by (2:5), ^P j b ^v (l) ((0; ^x 0 )) = (^v (l) ; ^~' ) j;(0;^k ) L 2 ( ^) j;0(0) ^' j;^k (^x 0 ); 8^x 0 2 [0; 1] n?1 ; ^P b j ^v (m) ((1; ^x 0 )) = ^k 2 ^K j ^k 2 ^K j (^v (m) ; ^~' j;(1;^k ) ) L 2 ( ^) j;1(1) ^' j;^k (^x 0 ); 8^x 0 2 [0; 1] n?1 ; with j;0 (0) = j;1 (1) = c 2 j=2 by (2:7) and (2:8). Thus, from (4:23), (4:24) and the triangle inequality, we get ^k 2 ^K j[(^v (l) ; ^~' j;(0;^k ) ) L 2 ( ^)? (^v(m) ; ^~' j;(1;^k ) ) L 2 ( ^) ] ^' j;^k L 2 ((0;1) n?1 ) < 2?sj [k^v (l) k H s ( ^) + k^v (m) k H s ( ^) ]:

18 18 C. Canuto, A. Tabacco, K. Urban Then, the result follows from (4:22) and the stability of the system f ^' j;^k g^k 2 ^K j L 2 ((0; 1) n?1 ). in the space Lemma 4.5 Let k 2 K j and set C(k) := f(l; m) 2 I(k) 2 l m is an (n? 1){faceg. For any i; i 0 2 I(k), jhv; ~' (i) j;k i i? hv; ~' (i0 ) j;k i i 0j 2 < ji(k)j (l;m)2c(k) jhv; ~' (l) j;k i l? hv; ~' (m) j;k i mj 2 : Proof. Under our assumptions on the there is a sequence of indices i 1 ; i 2 ; : : : ; i p 2 I(k) such that i 1 = i; i p = i 0 and for 1 q < iq iq+1 is a (n? 1){face. Then, the result follows by a telescoping argument. Lemma 4.6 Let i 2 f1; : : : ; Ng. Set D(i) := fi 0 i i 0 6= ;g. Assume that v 2 Hb 1 (; P) for some nonnegative s min(l; ). Then kr (i) vk j L < 2 (i) 2?sj kvk H s (i 0 ): i 0 2D(i) Proof. Let us rst assume s 1. By (4:21) and the L 2 {stability of the basis i j, we get kr (i) j vk2 L 2 (i) k2k j \@i k2k j \@i k2k j \@i ji(k)j?1 ji(k)j?1 ji(k)j i 0 2I(k) i 0 2I(k) (l;m)2c(k) [hv; ~' (i) i j;k i? hv; ~' (i0 ) i j;k i 0 ] 2 jhv; ~' (i) j;k i i? hv; ~' (i0 ) j;k i i 0j 2 jhv; ~' (l) j;k i l? hv; ~' (m) j;k i mj 2 by Lemma 4.5. Dene E(i) := f(l; m) 2 D(i) 2 :? l;m l \@ m is an (n?1){faceg. Recalling that ji(k)j < 1 and rearranging the last sum, we get kr (i) j vk2 L 2 (i) < (l;m)2e(i) k2k j \?l;m jhv; ~' (l) j;k i l? hv; ~' (m) j;k i mj 2 ; and the result for s 1 follows from Lemma 4.4. For s = 0, the result is a consequence of the L 2 {stability of Pj and P i j, whereas for 0 < s < 1 we conclude by interpolation. We are now ready to establish the Jackson inequality for P j. Theorem 4.7 Assume that v 2 Hb s (; P) for some nonnegative s min(l; ). Then, kv? P j vk L 2 () < 2?sj kvk H s b (;P): (4.25) Proof. In each i, we use the triangle inequality for v? Pj v = (v? P i j v) + R (i) j v and we conclude by (4:6) and Lemma 4.6.

19 The Wavelet Element Method, Part I 19 Remark 4.8 Note that (4:25) yields an optimal rate of decay of the approximation error even for those functions which are locally smooth in each subdomain, but not globally smooth in (i.e., functions which do not belong to H s ()). This feature turns out to be useful, for instance, in the numerical approximation of solutions of partial dierential equations. Finally, we consider the Bernstein inequality. Recalling the inclusion (4:13) and using (4:7), we easily get kvk H s b (;P) < 2 sj kvk L 2 (); 8v 2 V j (); 0 s : (4.26) This implies the possibility of characterizing the spaces Hb s (; P), as well as their duals, in terms of the L 2 {norms of the detail operators Q j := P j+1? Pj. The precise result will be given, after we provide a wavelet basis, see Theorem Biorthogonal wavelets on general domains We now construct biorthogonal complement spaces W j () and ~ Wj () (j j 0 ) such that V j+1 () = V j () W j (); V j ()? Wj ~ (); Vj+1 ~ () = Vj ~ () Wj ~ (); Vj ~ ()?W j (); (5.1) as well as the corresponding biorthogonal bases j and ~ j, where the orthogonality is to be understood with respect to h; i. Here we detail the construction for the primal functions only, i.e., for j, since the dual basis ~ j is built in a completely analogous fashion. To start with, let us dene a set of grid points by H j := K j+1 n K j = (see (4:14) and (4:8)). We shall associate to each h 2 H j a function j;h 2 V j+1 () and a function ~ j;h 2 Vj+1 ~ () such that j;h is orthogonal to Vj ~ (), ~ j;h is orthogonal to V j () and the biorthogonality conditions h j;h ; ~ j;h 0i = h;h 0 hold. Then, setting j := f j;h : h 2 H j g, ~ j := f ~ j;h : h 2 H j g, it will be clear that the spaces W j () := span j and Wj ~ () := span ~ j satisfy (5:1) (see Theorem 5.5). The construction will proceed as follows. Firstly, we build wavelets supported in the closure of only one subdomain. Next, we match wavelets and scaling functions across faces common to subdomains, starting from 0{faces and increasing the dimension of the face. Finally, the locally supported systems arising from the matching are biorthogonalized. Let us x h 2 H j. By denition, there exists i 2 f1; : : : ; Ng and ^h = ^h (i) 2 r b( i) j such that h = F i (^h), i.e., ^h is the corresponding grid point on the reference domain. Recalling the denition of internal grid points on the reference interval [0; 1] (see (2:9)), let p = p(^h) 2 f0; : : : ; ng be the number of components ^h l of ^h belonging to int j [ r int j. Furthermore, let us dene the auxiliary point h := F i (^h ) 2 K j+1 by setting, for 1 l n, 8 >< ^h l ; if ^h l is internal; ^h l := 0; if >: ^h l 2 f0; j;1 g; (5.2) 1; if ^h l 2 f1; j;mj g: N[ i=1 H i j

20 20 C. Canuto, A. Tabacco, K. Urban Figure 1: Grid points around a cross point C surrounded by the subdomains i, i = 1; : : : ; 5. The mapping h 7! h will be denoted by F. To be precise, we should write F i, but since F i (h) = F i 0(h) if h i i 0, we are allowed to drop the index of the subdomain and to consider F as a mapping from H j to K j+1. It will be useful to consider the set H j (h ) := fh 2 H j : F(h) = h g = F?1 (h ): (5.3) The simplest situation occurs when p = n. In this case, F(h) = h 2 i and indeed H j (h) = fhg; moreover, the wavelet (i) 2 j;h W j( i ) (dened in (4:9)) vanishes identically i ; thus, we associate to h the function of V j+1 () j;h(x) := ( (i) j;h (x); if x 2 i; 0; elsewhere. (5.4) If p < n, then h belongs to a p{face of i. Two situations may occur. If h does not belong to i 0 for i 0 6= i, then it lies on the boundary of and (i) j;h vanishes i Thus, we associate to h the wavelet j;h dened as in (5:4). Otherwise, h belongs to a face common to at least two subdomains, and we have to enforce a matching. In the sequel, we construct a set of linearly independent functions in V j+1 () which will be associated to the set H j (h ). 5.1 Matching at a cross point Let us start with the case in which h =: C is a cross point, i.e., a 0{face common to N C subdomains, that we assume to be (re-)labeled by 1 ; : : : ; NC. Let us rst consider the case C 2 (see Figure 1), next we shall indicate the modications when C Internal cross points For each i ; i 2 f1; : : : ; N C g, there are exactly 2 n?1 points h 2 H i j such that h = C. Including C itself, we have 2 n points of the form h = F i (^h), where ^h = (^h l ) l is such that each component

21 The Wavelet Element Method, Part I 21 ^h l ranges either in the set f0; j;1 g or in the set f j;mj ; 1g. This set of points can be identied with the set E n = f0; 1g n by the mapping ( h 7! 0; if ^hl 2 f0; 1g; e = (e l ) l ; with e l := 1; if ^h l 2 f j;1 ; j;mj g. In turns, the vector e is associated with the function in V j+1 ( i ) where ^x = G i (x) and (i) e (x) = ^(i) e (^x) := ny l=1 # (i) l (^x l ); (5.5) # (i) j;0 ; if e l = 0, l := j;j;1 ; if e l = 1, if (G i (C)) l = 0; (i.e., if we are in a neighborhood of the left hand side of the interval [0; 1]), or # (i) l := ( j;1 ; if e l = 0, j;j;mj ; if e l = 1, if (G i (C)) l = 1: The set V C j+1( i ) := spanf (i) e : e 2 E n g (5.6) is a subset of V j+1 ( i ) of dimension 2 n. An element v (i) 2 V C ( j+1 i) is uniquely determined by the column vector (i) = ( (i) e ) e2e n by the relation v (i) = e e (i) : (5.7) e2e n (i) Considering all the subdomains meeting at C, we have 2 n N C free coecients, which form the vector = ( (i) ) 1iNC. Continuity the space Since we are interested in continuous wavelets across interelements, we introduce V C j+1() := fv 2 C 0 () : v j i 2 V C j+1( i ); if i 2 f1; : : : ; N C g; v j i 0 elsewhereg: By the representation (5:7), an element v 2 V C j+1 () is associated to a vector, which belongs to the kernel of a certain matrix C representing an appropriate set of continuity conditions. We are now going to show a particular choice of such conditions and to construct the corresponding matrix. The perhaps most natural approach would be to consider any (n? 1){face? i;i 0 common to with i; i 0 2 f1; : : : ; N C g, and to impose the matching between the 0). (Note indeed that functions belonging to these spaces identically vanish on all (n? 1){faces which do not contain C.) This would lead to 2 n?1 conditions, linearly independent with respect to each other. However, certain matching conditions corresponding to dierent (n?1){faces are linearly dependent, and it is not obvious how to select a maximal set of linearly independent conditions. To avoid this problem, we consider all the p{faces, with 0 p n? 1, which contain the cross point C, and we enforce two subdomains i and i 0 restriction to? i;i 0 of functions v (i) 2 V C j+1 ( i) and v (i0 ) 2 V C j+1 ( i

22 22 C. Canuto, A. Tabacco, K. Urban one suitable matching condition along each face. We prove that all these conditions are linearly independent, and that they are equivalent to the matching conditions along all the (n? 1){faces containing C. To be precise, let be a p{face containing C, and let i be a subdomain having as a face. Then, = F i (^), where ^ is dened as in (4:1) by a set L^ of frozen coordinates and corresponding values. The following notation will be useful. Given t 2 [0; 1] n, set _t = _D^ t := (t l ) l =2L^ 2 [0; 1] p (5.8) (i.e., we delete the components of t corresponding to the frozen coordinates of ^), and t = D^ t := (t l ) l2l^ 2 [0; 1] n?p : Conversely, given _t 2 [0; 1] p and t 2 [0; 1] n?p, let t = R^ (_t; t) 2 [0; 1] n (5.9) be the unique vector such that _t = _D^ t and t = D^ t (i.e., t is reconstructed from _t and t according to the position of the frozen coordinates of ^). Moreover, given _e 2 E p and ^y = (^y l ) l 2 [0; 1] p, we dene ^(i) _e (^y) as in (5:5) with n replaced by p. Finally, recalling conditions (2:7) and (2:8), let us denote by j := j;0 (0) = j;j;1 (0) = j;1 (1) = j;j;mj (1); ~ j := ~ j;0 (0) = ~ j;j;1 (0) = ~ (5.10) j;1 (1) = ~ j;j;mj (1) the common value of the scaling and wavelet functions at the end points of the interval [0; 1]. Given any v (i) 2 V C j+1 ( i), represented as in (5:7), we have v (i) j(x) = e e (i) j (x) Let i 0 we have as above e2e n (i) = n?p j = n?p j =: n?p j (i) ^(i) e e2e n _e2e p b (i) _e _e2e p _D^ e ( _D^ ^x); (i) e e2e n?p ^(i) _e (^y): 1 ^x = G i (x); A ^(i) _e (^y); ^y = _D^ (^x); e = R^ ( _e; e); (5.11) be another subdomain having as a face, and let = F i 0(^ 0 ). Given v (i0 ) 2 V C j+1 ( i 0), v (i0 ) j (x) = n?p j b (i0 ) _e _e2e p ^(i0 ) _e (^y 0 ); (5.12) P where b (i0 ) _e = e2e n?p ) (i0 e with e 0 = 0 R^ 0( _e; e), and ^y 0 = _D^ 0(G i 0(x)). Now, by Hypothesis (4:2), the mapping T : ^y 7! ^y 0 is a composition of reections and permutations of coordinates, say T = R P. Therefore, using (2:4) if there are reections, ^(i0 ) _e (^y 0 ) = ^(i0 ) _e (T ^y) = = = py l=1 py m=1 # (i0 ) l ((R^y) P (l) ) = py l=1 py m=1 # (i0 ) l ((T ^y) l ) # (i) P?1 (m) (^y m) = ^(i) P?1 _e (^y): # (i) P?1 (m) ((R^y) m)

23 The Wavelet Element Method, Part I 23 Hence, (5:12) becomes v (i0 ) j (x) = n?p j b (i0) _e _e2e p ^(i) P?1 _e (^y) = n?p j b (i0) ^(i) P _e _e _e2e p By the linear independence of the functions f ^(i) _e : _e 2 E p g, the matching condition v (i) j(x) = v (i0 ) j (x) is equivalent to the 2 p conditions (^y): b (i) _e = b (i0 ) P _e ; 8 _e 2 Ep : (5.13) We choose to enforce one particular combination of these conditions. This combination is uniquely associated to the face and the couple of subdomains i and i 0. For any e 2 E q, q 1, let us set sgn e := (?1) jej ; jej := i.e., sgn e is +1 or?1, depending on the parity of the number of 1's in e; we also set sgn e := 1 when q = 0. Then, we require that (sgn _e) b (i) _e = (sgn _e) b (i0) P _e : (5.14) _e2e p _e2e p By observing that sgn P _e = sgn _e for any permutation P of components, (5:14) can be equivalently written as (sgn _e) b (i) _e = (sgn _e) b (i0) _e : (5.15) _e2e p _e2e p We want to express this condition in terms of the coecients of the expansions (5:7) for v (i) and v (i0 ). To this end, let us introduce the row vector Then, (5:15) can be written as q l=1 e l ; c (i) := (sgn _D^ e) e2e n 2 f?1; 1g 2n : (5.16) c (i) (i) = c (i0 ) (i0) : The following lemma will be crucial in the sequel. Lemma 5.1 Let and be two dierent faces of the same subdomain i containing C. Then c (i) (c (i) ) t = 0: (5.17) Proof. For convenience, we drop the index (i) throughout the proof. Let us assume that is a p{face, and is a q{face with p q. Then c (c) t = (sgn _D^ e) (sgn _D^ e) e2e n = (sgn _D^ e) (sgn _D^ e); with e = R( _e; e); _e2e p e2e n?p = sgn _e sgn _D^ e: _e2e p e2e n?p