Representing the Geometry of Domains by Wavelets with Applications to Partial Diierential Equations

1 INTRODUCTION The recently introduced compactly supported wavelets of Daubechies have proven to be very useful in various aspects of signal processing, notably in image compression (see, e.g., 23]). The fundamental idea in the application to image compression was to take a digitized image and to use wavelets to provide a multiscale representation of it, and to then discard some information at some scales, and leave information more intact at other scales. The Daubechies wavelets have diierentiability properties in addition to their compact support and orthogonality properties. A number of authors have used such wavelet systems for solving various problems in diierential and An additional point of view for approaching boundary value problems in partial diierential equations was introduced in 18], in which the boundary, the boundary data, and the unknown solution (of a boundary value problem) on the interior of a domain are all uniformly represented in terms of compactly supported wavelet functions in an extrinsic ambient Euclidean space. In this paper we describe multiscale representations of domains and their boundaries and obtain multiscale representations of some of the basic elements of geometric calculus (line integrals, surface measures, etc.) which are then in turn useful for speciic numerical calculations in problems in approximate solutions of diierential equations. This is similar to the spectral method in solving diierential equations (essentially using an orthonormal expansion), but here we use the localization property of wavelets to extend this orthonormal representation to boundary data and geometric boundaries. In this paper we want to indicate how to carry this out. We illustrate this principle for the Dirichlet problem for a classical elliptic diierential equation: ?u + u = f in ; uj @ = g; (1:1) where is a bounded domain in R n , with a boundary @ satisfying a suitable regularity condition and where f and g are suitable given functions. The basic principle divides into several basic points: 1. Represent the geometric region for the boundary value problem in terms of wavelets. For this we use the characteristic function of the domain and let r represent its boundary as a measure on R n. Both of these can be represented in terms of a wavelet series, using diierentiable wavelets and connection coeecients (the latter are useful for calculating derivatives of wavelet expansions, see 11]). 2. Represent the functions deened on the boundary and on the interior of the region in terms …