Exponential convergence and <i>H‐c</i> multiquadric collocation method for partial differential equations

TLDR

The radial basis function collocation method interpolates and collocates PDE solutions using global shape functions and is a truly meshless approach compared to other meshless or element‑free finite element methods. The study aims to establish an exponential error estimate for the RBF collocation method through numerical experiments. The authors employ the c‑scheme, increasing the shape parameter instead of refining mesh size, and use residual error as an indicator to optimize c without extra computational cost. Numerical experiments confirm the error behaves as ϵ ∼ O(λ √ c̄ h) with 0 < λ < 1. © 2003 Wiley Periodicals, Inc., Numer Methods Partial Differential Eq 19: 571–594.

Abstract

Abstract 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 ϵ ∼ O (λ √ c̄h ) where 0 &lt; λ &lt; 1. We also propose the use of residual error as an error indicator to optimize the selection of c . © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 571–594, 2003