Wavelets on manifolds - I: Construction and domain decomposition

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 others working in a similar setting. The present construction of wavelets is closely intertwined with a suitable characterization of function spaces over such a manifold in terms of product spaces, where each factor is a corresponding local function space subject to certain boundary conditions. Wavelet bases for each factor can be obtained as parametric liftings from bases on the standard cube satisfying appropriate boundary conditions. The use of such bases for the discretization of operator equations leads in a natural way to a conceptually new domain decomposition method. It is shown to exhibit the same favorable convergence properties for a wide range of elliptic operator equations covering, in particular, also operators of nonpositive order. In this paper we address all three issues, namely, the characterization of function spaces which is intimately intertwined with the construction of the wavelets, their relevance with regard to matrix compression and preconditioning as well as the domain decomposition aspect.