Solving Partial Diierential Equations by Collocation with Radial Basis Functions

Motivated by 5] we describe a method related to scattered Hermite interpolation for which the solution of elliptic partial diierential equations by collocation is well-posed. We compare the method of 5] with our method. x1. Introduction In this paper we discuss the numerical solution of elliptic partial diierential equations using a collocation approach based on radial basis functions. To make the discussion transparent we will focus on the case of a time independent linear elliptic partial diierential equation in IR 2. In the following we assume we are given a set of nodes = f ~ 1 ; : : :; ~ N g IR d , along with a continuous function ' : 0; 1) ! IR. We then refer to ~ x 7 ! '(k~ x? ~ k k 2), ~ x 2 IR d , k 2 f1; : : :; Ng, as radial basis functions centered at ~ k. Some of the most commonly used radial basis functions are the (reciprocal) multiquadrics '(r) = (r 2 + c 2) 1=2 , the Gaussians '(r) = e ?cr 2 , and the thin plate splines '(r) = r 2 log r in IR 2. Here we let r = k~ x ? ~ k k, and c > 0 is a parameter. Some of the advantages of radial basis functions are their insensitivity to the spatial dimension d, which makes the implementation of this method in higher dimensions much simpler than, e.g., nite elements. Another useful feature of radial basis functions is their radial symmetry and invariance under Euclidean transformations. Furthermore, in the context of scattered data interpolation it is known that some radial basis functions have spectral convergence orders (e.g., (reciprocal) multiquadrics, Gaussians). This should also be evident in some form when using them for collocation. There are also some well-known disadvantages. Radial basis functions are generally globally supported and poorly conditioned. One usually tries to remedy these problems by localization, preconditioning, and ne tuning of the parameter c. ISBN 1-xxxxx-xxx-x. All rights of reproduction in any form reserved.