Spatially adaptive multiwavelet representations on unstructured grids with applications to multidimensional computational modeling

Abstract

In this thesis, we develop wavelet surface wavelet representations for complex surfaces, with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. However, we further extend the construction of surface wavelets and prove the existence of a large class of multiwavelets in Rn with vanishing moments around corners that are well suited for complex geometries. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modeling. This motivates the study of surface wavelets as an efficient representation for the modeling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, cast in the integral form. We analyze and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties.

(cont.) We show both theoretically and experimentally that an O(h2/n) convergence rate, hn being the mesh size, can be obtained by retaining only O((logN)7/2 N) entries in the discrete operator matrix, where N is the number of unknowns. Moreover our theoretical proof of accuracy vs compression is applicable to a large class of Calderón-Zygmund integral operators. In principle, this convergence analysis may be extended to higher order wavelets with greater vanishing moment. This results in higher convergence and greater compression.