Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Saturday, January 25th, 1997

Biorthogonal Wavelet Bases on Domains and Manifolds

Multiscale bases like wavelets offer an versatile and efficient tool for adaptive approximation. For example, they give rise to sparse dicretization for integral equations and promise various application in the numerical solution for partial differential equations. A central problem in this regard is the adaptation of multiscale bases to bounded domains and arbitrary manifolds or surfaces and the incorporation of boundary conditions.

We construct biorthogonal wavelet bases on domains and manifolds \Gamma, which are given by the union of parametric images of n-dimensional cubes. The order of approximation d and the number of vanishing moments d^* of the primal wavelets can be chosen in a way that d+d^* even. The primal spaces are spline spaces and the regularity of the dual functions is well controlled. Biorthogonality is realized by a modified inner product and the primal as well as the dual functions are globally continuous. Homogeneous Dirichlet boundary conditions can be posed along the boundary of the domain or manifold or along a part of the boundary. A major tool for construction is the stable completion. The construction is based biorthogonal wavelets bases on an intervall and tensor product construction for cubes. Afterwards the functions are lifted to the corresponding parametric patches through the parametrization. One can pose either homogeneous boundary conditions or no boundary condition on each face of the cube. Particular attention is paid for gluing the basis functions continuously together.

These multiscale basis functions are appropriate for a conform Galerkin discretization of second order partial differential operators and of (singular) integral operators of nonegative order. They are only suboptimal for operators of order -1. In order to remedy this situation we present a new domain decomposition approach based on multiscale extension operators. With this method one can treat also operators of negative and higher order as well.

Numerical examples are concerning matrix compression for boundary integral equations.