Hierarchical Solution of PDEs Using Wavelets.
Abstract
In engineering problems, we often require a quick rough estimate of the solution at the preliminary stage, which may later be refined as the design or investigation progresses. The multiresolution properties of wavelets suggest that is possible to obtain an initial coarse description of the solution with little computational effort and then successively refine the solution in regions of interest with a minimum of extra effort. The problem of successive refinement is one of the main drawbacks of the finite element method. This paper demonstrates how a hierarchy of solutions to a PDE can be obtained by using Mallat's multiresolution transform in conjunction with the wavelet-Galerkin method. This approach provides a rational means to trade off accuracy for solution speed. In contrast to the example of Beylkin et. al. where the discrete wavelet transform is applied to the matrix differential operator d/dx, we decompose the inverse of the differential operator matrix. We note that the structure of the inverse matrix is particularly suitable for developing hierarchical solutions.
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 01, 1993
- Accession Number
- ADP009085
Entities
People
- John R. Williams
- Kevin Amaratunga
Organizations
- Massachusetts Institute of Technology