M ar 1 99 8 The solution of multi - scale partial differential equations using wavelets

Wavelets are a powerful new mathematical tool which offers the possibility to treat in a natural way quantities characterized by several length scales. In this article we will show how wavelets can be used to solve partial differential equations which are characterized by widely varying length scales and which are therefore hardly accessible by other numerical methods. The standard way to solve partial differential equations is to express the solution as a linear combination of so-called basis functions. These basis functions can for instance be plane waves, Gaussians or finite elements. Having discretized the differential equation in this way makes it amenable to a numerical solution. Wavelets are just another basis set which however offers considerable advantages over alternative basis sets. Its main advantages are: