An Investigation of Wavelet Bases for Grid-Based Multi-Scale Simulations Final Report

The research summarized in this report is the result of a two-year effort that has focused on evaluating the viability of wavelet bases for the solution of partial differential equations. The primary objective for this work has been to establish a foundation for hierarchical/wavelet simulation methods based upon numerical performance, computational efficiency, and the ability to exploit the hierarchical adaptive nature of wavelets. This work has demonstrated that hierarchical bases can be effective for problems with a dominant elliptic character. However, the strict enforcement of orthogonality was found to be less desirable than weaker semi-orthogonality or bi-orthogonality for solving partial differential equations. This conclusion has led to the development of a multi-scale linear finite element based on a hierarchical change of basis. The reproducing kernel particle method has been found to yield extremely accurate phase characteristics for hyperbolic problems while providing a convenient framework for multi-scale analyses