Multiscale Wendland radial basis functions and applications to solving partial differential equations

Abstract

Meshfree methods, which use linear combinations of radial basis functions (RBFs) to construct approximations, have become popular for the numerical solution of partial differential equations (PDEs). The Wendland functions are a class of compactly supported, piecewise polynomial RBFs which are important as they use the minimum degree polynomial for a specified smoothness and their compact support leads to sparse linear systems. A practical issue is the choice of scale to use for the RBFs. A small scale will lead to a sparser and better conditioned linear system, but at the price of poor approximation power. Conversely, a large scale will have better approximation power but at the price of an ill-conditioned linear system. We firstly consider a generalisation of the Wendland functions, which allows greater freedom in the choice of parameters, and give sufficient and necessary conditions for these functions to be positive definite, as well as classifying the native spaces generated. We give closed form representations for and properties of the Wendland functions and their Fourier transforms. By an appropriate choice of scaling, we investigate the behaviour of the Wendland functions as the smoothness parameter goes to infinity. This provides insights into the selection of the parameters of the Wendland functions. We then consider multiscale algorithms for the numerical solution of PDEs. These construct the approximations over several levels, each level using a Wendland RBF with a different scaling factor. We present a theoretical and practical analysis of two multiscale algorithms for Galerkin approximation of elliptic PDEs on bounded domains, including results on convergence and condition numbers. Convergence is investigated in terms of the mesh norm and the angles between subspaces. The issue of the supports of the RBFs overlapping the boundaries is also considered in our stability analysis. Finally we consider a multiscale algorithm for collocation approximation of elliptic PDEs and the Stokes problem on a bounded domain. We provide results on convergence and condition numbers. For the Stokes problem, we use a divergence free RBF constructed from the Wendland functions, since the Wendland functions are not divergence free.