Wavelets on Regular Surfaces Generated by Stokes Potentials

Abstract

By means of the limit and jump relations of potential theory with respect to the Stokes equations the framework of a tensorial wavelet approach on a regular (Lyapunov-) surface is established. The setup of a multiresolution analysis is defined by interpreting the kernel functions of the limit and jump integral operators as scaling functions on the regular surfaces. The distance of the parallel surface to the surface under consideration thereby represents the scale level in the scaling function. Tensorial scaling functions and wavelets show space localizing properties. Thus, they can be used to represent vector fields locally on a regular surface. Furthermore, these functions can be used as ansatz functions for the discretization of Fredholm integral equations of the second kind which result from the Stokes boundary-value problems with respect to a regular surface. By this, scaling functions and wavelets enable us to give a multiscale representation of the solution of the Stokes problem. This representation will be demonstrated in a concrete example.