An adaptive‐hybrid meshfree approximation method

Abstract

SUMMARYIt is now commonly agreed that the global radial basis functions (GRBF) method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of ill‐conditioning and the high computational cost for solving dense matrix systems. We previously proposed different variants of adaptive methods for selecting proper trial subspaces so that the instability caused by inappropriately shaped parameters were minimized. In contrast, the compactly supported radial basis functions (CSRBF) are more relaxing on the smoothness requirements. By settling with the algebraic order of convergence only, the CSRBF method, provided the support radii are properly chosen, can approximate functions with less smoothness. The reality is that end users must know the functions to be approximated a priori to decide which method to be used; this is not practical if one is solving a time‐evolving partial differential equation. The solution could be smooth at the beginning but the formation of shocks may come later. In this paper, we propose a hybrid algorithm making use of both GRBF and CSRBF with other developed techniques for meshfree approximation with minimal fine tuning. The first contribution here is an adaptive node refinement scheme. Second, we apply the GRBFs (with adaptive subspace selection) on the adaptively generated data sites, and lastly, the CSRBF (with adaptive support selection) that can be used as a blackbox algorithm for robust approximations to a wider class of functions and for solving PDEs. Copyright © 2011 John Wiley & Sons, Ltd.