Published: 2001
Total Pages:
Get eBook
Constructing numerical schemes which are both adaptive and suitable for parallel architectures is very challenging. The challenge lies in the need to maintain a balanced load across the processing elements using a method that is both efficient and scalable. Here we propose a method which is adaptive, load balanced, absolutely efficient and scalable offering significant speedup over lower order adaptive schemes. The ability of wavelets to accurately and efficiently represent functions with localized features has spawned intensive research into applying wavelets for the solution of partial differential equations with the promise of significantly reducing the necessary computational effort and memory requirements. Traditionally, this effort has been centered around using wavelets as an orthogonal and complete basis, spanning a space in which to seek approximate solutions satisfying the equation in a Galerkin sense. Besides from the well known difficulties associated with such an approach for non-linear problems, one is also faced with the problem of dealing with non-trivial boundary conditions in an accurate and stable manner. Such restrictions on the applicability of wavelet based methods for the solution of problems of more general interest have, in recent years, induced significant interest into grid-based collocation wavelet methods, with various different approaches being taken. The formulation and implementation of multi-dimensional pure wavelet collocation methods, however, remains a challenging task and many issues require attention. In the present work we take a somewhat different approach to arrive at a grid based method utilizing the unique properties of wavelets. Rather than using the wavelets as a basis, we utilize the ability of wavelets to not only detect the existence of high-frequency information but also to supply information about the spatial location of such strongly inhomogeneous regions.
