Nonsymmetric Multilevel RBF Collocation within an Operator Newton Framework for Nonlinear PDEs

We present a multilevel algorithm based on nonsymmetric collocation with compactly supported radial basis functions for the solution of nonlinear partial diierential equations. At the operator level this algorithm can be viewed as a Newton method. Details of the algorithm as well as numerical experiments are discussed. x1. Introduction Radial basis function (RBF) methods have been studied by various authors over the past two decades. As a general introduction we suggest the recent textbook 1]. Most of the research (theoretical and applied) has focussed on the use of RBFs in approximation theory (i.e., scattered data approximation). More recently, however, some authors have also considered using RBFs for the numerical solution of partial diierential equations (PDEs). An overview of some of these approaches is given in 4]. In this paper we will show how RBF collocation (brieey explained in Sect. 3) can be applied to the solution of nonlinear PDEs. More speciically, we will begin in Sect. 2 by reviewing an operator Newton framework discussed in detail in 7], and then couple this approach with compactly supported RBFs (brieey discussed in Sect. 4). A short motivation of why the kind of multilevel/Newton framework suggested here is the natural environment in which to employ compactly supported RBFs follows in Sect. 5. Our suggested approach leads to an algorithm whose details we discuss in Sect. 6, and whose performance we illustrate in Sect. 7. Some closing remarks are provided in Sect. 8. ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.