Wavelets on manifolds and domain decomposition

The potential of wavelets as a discretization tool hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet coefficients as well as on certain cancellation properties. Both features are crucial for the construction of optimal preconditioners, for matrix compression and the corresponding sparse representations of functions and operators and the disign and analysis of adaptive solvers. So far the availability of such bases is confined to very simple domain geometries.

This talk is concerned with concepts that aim at expanding the applicability of current wavelets schemes. The key 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 is based on the 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 whose convergence properties are well understood also for operators of nonpositive order.