Numerical Stability on Local Integral Methods using RBF-QR

Abstract

Many local integral methods are based on an integral formulation over small and heavilly overlapping stencils with local RBF interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix -- when the RBF shape parameter tends to zero corresponding to best accuracy -- is a serious task. Several stabilizing methods have been developed to deal with this near flat RBFs. The inclusion of the RBF-QR technique in the process of approximating in local integral methods makes possible to avoid this problem and stabilize the numerical error. In this paper we combine this technique in a local integral method and present accuracy results for Poisson, convection-difussion and thermal boundary layer PDEs.