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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.

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Ue have used the =ultiquadric (MO) approximation scheme foc the solution of elliptic partial di_ferential equations with Oirichlet and/or Neumann boundary conditions. The scheme has the advantage to use tha data points in arbitrary locations with an arbitrary ordering. Two di==ensional Laplace, Poisson and 8iharmonic equations describing the vario,_s physical processes, have been taken as the test examples. The agreement is ¢ound to be very good between the comp,ated and e: