The Numerical Performance of Wavelets and Reproducing Kernels for PDE's

The results presented here constitute a brief summary of an on-going multi-year e ort to investigate hierarchical/wavelet bases for solving PDE's and establish a rigorous foundation for these methods. A new, hierarchical, wavelet-Galerkin solution strategy based upon the Donovan-GeronimoHardin-Massopust (DGHM) compactly-supported multi-wavelet is presented for elliptic partial di erential equations. This multi-scale wavelet-Galerkin method uses a wavelet transform to yield nearly mesh independent condition numbers for elliptic problems as opposed to the multi-scaling functions that yield condition numbers which increase as the square of the mesh size. In addition, the results of von Neumann analyses for the DGHM multi-wavelet element and the Reproducing Kernel Particle Method (RKPM) are presented for model hyperbolic partial di erential equations. RKPM exhibits excellent dispersion characteristics using a consistent mass matrix with the proper choice of re nement parameter and mass matrix formulation. In comparison, the wavelet-Galerkin formulation using the DGHM element delivers a frequency response comparable to a Bubnov-Galerkin formulation with a quadratic element.