Spatially Adapted Multiwavelets and Sparse Representation of Integral Equations on General Geometries

In this paper we develop irregular wavelet representations for complex domains with the goal of demonstrating their potential for three-dimensional (3D) scientific and engineering computing applications. We show existence and construction of a large class of continuous spatially adapted multiwavelets in $R^{\eta}$ with vanishing moments over complex geometries. These wavelets share all of the major advantages of conventional wavelets in that they provide an analytical tool for studying data, functions, and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, spatially adapted multiwavelets allow fast transform, localization, and decorrelation on complex meshes, such as those encountered in finite element modeling. We show how these new constructions can be applied to partial differential equations cast in the integral form. We implement the wavelet approach for several model two-dimensional (2D) and 3D potential problems. It is shown that the optimal convergence rate is achieved, with only ${\cal O}( N ( {{\rm log}N} )^{\alpha} )$ entries of the discrete operator matrix, where $\alpha$ is a small number and N is the number of unknowns.