this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.

This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit d-cube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudo-differential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions. Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods fo...

The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on part of the boundary. The essential point is that the primal and dual wavelets satisfy certain corresponding complementary boundary conditions. These results form the key ingredients of the construction of wavelet bases on manifolds [DS2] that have been developed for the treatment of operator equations of positive and negative order. Key Words: Topological isomorphisms, Sobolev and Besov spaces, biorthogonal wavelet bases, moment conditions, complementary boundary conditions. AMS Subject Classification: 46A20, 46E39, 46B15, 1 Introduction A number of recent investigations [CTU, DS1, DSt] aimed at extending the applicability of wavelet methods for the numerical treatment of operator equations to problems involving realistic domain geometries. In spite of the extremely promising pote...

This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions the best possible approximation rate, which a function can have for a given number of degrees of freedom, is characterized in terms of its regularity in a certain scale of Besov spaces. Therefore, after demonstrating the gain of nonlinear approximation over linear approximation measured in a Sobolev scale, we review some recent results on the Sobolev and Besov regularity of solutions to elliptic boundary value 1 problems. On the other hand, nonlinear approximation requires information that is generally not available in practice. Instead one has to resort to the concept of adaptive approximation. We briefly summarize some recent results on wavelet based adaptive schemes for elliptic operator equations. In co...

In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional aposteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.

In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly non-uniform meshes on n- dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and the wavelets can, in principal, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used e.g. for constructing an optimal solver of discretized H s -elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet ba...

In this paper an adaptive wavelet scheme for saddle point problems is developed and analysed. Under the assumption that the underlying continuous problem satisfies the inf-sup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the best wavelet N-term approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the Ladyshenskaja-Babuska-Brezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.

In "Adaptive Wavelet Methods II --...

We establish multiresolution norm equivalences in weighted spaces w ((0, 1)) with possibly singular weight functions w(x) 0 in (0, 1).