Solution of Variational Problems 363 x 3

In this paper we investigate the application of radial basis functions to solve variational problems as they appear in the weak formulation of elliptic partial diierential equations. Two algorithms based on a multilevel scheme are introduced. Their numerical advantages and disadvantages are discussed and demonstrated on an example. x1. Introduction The classical Galerkin approach for the solution of elliptic partial diierential equations leads to a variational problem, which can be solved by discretization. This is the way nite elements treat this kind of problems in a very successful way. The aim of this paper is to give rst ideas how radial basis functions can be used in this setting. There are several good reasons to investigate this meshless method (cf. 1] for an overview on general meshless methods and the literature in 3] for collocation-based methods), e.g., the independence of space dimension (in contrast to classical nite element methods) and the independence of the underlying grid. The latter might be interesting in time dependent problems with moving boundaries, where classical nite element methods spend a lot of time for generating and adapting the mesh. The paper 9] deals with radial basis functions in the context of Galerkin approximation. But in contrast to its theoretical approach, we now want to investigate the numerical side of the problem. Therefore, we concentrate on the Helmholtz equation with natural boundary conditions: ?u + u = f in ; @ @ u = 0 on @: Here IR d is a bounded domain with a suuciently smooth boundary @ and f is a given function. The outer unit normal vector is denoted by. See also 5] for the connection of radial functions and the Helmholtz equation. All rights of reproduction in any form reserved.