Wavelets on manifolds - I: Construction and domain decomposition

The potential of wavelets as a discretization tool for the numerical treatm ent of operator equations hinges on the validity of norm equivalences for B esov or Sobolev spaces in terms of weighted sequence norms of wavelet expan sion coefficients and on certain cancellation properties. These features ar e crucial for the construction of optimal preconditioners, for matrix compr ession based on sparse representations of functions and operators as well a s for the design and analysis of adaptive solvers. However, for realistic d omain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts tha t 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 parametr ic images of the standard cube. The approach considered here is conceptuall y different though from others working in a similar setting. The present co nstruction of wavelets is closely intertwined with a suitable characterizat ion of function spaces over such a manifold in terms of product spaces, whe re each factor is a corresponding local function space subject to certain b oundary conditions. Wavelet bases for each factor can be obtained as parame tric liftings from bases on the standard cube satisfying appropriate bounda ry conditions. The use of such bases for the discretization of operator equ ations leads in a natural way to a conceptually new domain decomposition me thod. It is shown to exhibit the same favorable convergence properties for a wide range of elliptic operator equations covering, in particular, also o perators of nonpositive order. In this paper we address all three issues, n amely, the characterization of function spaces which is intimately intertwi ned with the construction of the wavelets, their relevance with regard to m atrix compression and preconditioning as well as the domain decomposition a spect.