Wavelet - Galerkin solutions for one dimensional partial di erential equations

In this paper we describe how wavelets may be used to solve partial diierential equations. These problems are currently solved by techniques such as nite diierences, nite elements and multi-grid. The wavelet method, however, ooers several advantages over traditional methods. Wavelets have the ability to represent functions at diierent levels of resolution, thereby providing a logical means of developing a hierarchy of solutions. Furthermore, compactly supported wavelets (such as those due to Daubechies 1]) are localized in space, which means that the solution can be reened in regions of high gradient, e.g. stress concentrations, without having to regenerate the mesh for the entire problem. In order to demonstrate the wavelet technique, we consider the one dimensional counterpart of Helmholtz's equation. By comparison with a simple nite diierence solution to this problem with periodic boundary conditions we show how a wavelet technique may be eeciently developed. Dirichlet boundary conditions are then imposed, using the capacitance matrix method decribed by Proskurowski and Widlund 2] and others. The convergence of the wavelet solutions are examined and they are found to compare extremely favourably to the nite diierence solutions. Preliminary investigations also indicate that the wavelet technique is a strong contender to the nite element method, at least for problems with simple geometries.