The radial basis function (RBF) collocation method uses global shape functions to interpolate and collocate the approximate solution of PDEs. It is a truly meshless method as compared to some of the so-called meshless or element-free finite element methods. For the multiquadric and Gaussian RBFs, there are two ways to make the solution converge-either by refining the mesh size h, or by increasing the shape parameter c. While the h-scheme requires the increase of computational cost, the c-scheme is performed without extra effort. In this paper we establish by numerical experiment the exponential error estimate ⑀ ϳ O( ͌c/h ), where 0 Ͻ Ͻ 1. We also propose the use of residual error as an error indicator to optimize the selection of c.

No abstract

No abstract

No abstract

We obtain explicit analytical particular solutions for Helmholtz-type operators, using higher order splines. These results generalize those in Golberg, Chen and Rashed (1998) and Chen and Rashed (1998) for thin plate splines. This enables one to substantially improve the accuracy of algorithms for solving boundary value problems for Helmholtz-type equations.