Wavelets on Domains and Surfaces

Wavelet discretizations of boundary value problems exhibit several advantages, like well-conditioning, compressibility, or adaptivity. Particularly our wavelet Galerkin scheme solves boundary integral equations in linear complexity. To that end, wavelet bases are required which provide sufficiently many vanishing moments. Our realization is based on biorthogonal spline wavelets derived from the multiresolution developed by Cohen, Daubechies and Feauveau.

   
Parametric surface representations with mesh on level 3

The domain or surface is subdivided into smooth four-sided patches which are images of the unit square under smooth diffeomorphisms. Such parametric surface representations can directly be obtained from computer aided design (CAD) and are recently studied in the context of isogeometric analysis. The wavelets are constructed by lifting wavelets from the unit square to the surface via parametrization. Gluing across the patch boundaries yields globally continuous wavelet bases.

   
primal scaling function at a degenerated vertex



     dual scaling function at a degenerated vertex



   
primal wavelet near an interface     dual wavelet near an interface


Selected Publications

  1. H. Harbrecht and R. Schneider. Biorthogonal wavelet bases for the boundary element method. Math. Nachr., 269-270:167-188, 2004.

  2. H. Harbrecht and R. Stevenson. Wavelets with patchwise cancellation properties. Math. Comput., 75(256):1871-1889, 2006.

  3. H. Harbrecht and M. Randrianarivony. From Computer Aided Design to Wavelet BEM. Comput. Vis. Sci., 13(2):69-82, 2010.