A comparison of efficiency and error convergence of multiquadric collocation method and finite element method

Abstract

In this paper, we demonstrate the efficiency and accuracy of the multiquadric collocation method as compared to the finite element method. When used as interpolants, the multiquadric (MQ) radial basis function has the property of exponential convergence with respect to a shape parameter, according to a proof by Madych and Nelson. Although the optimal choice of shape parameter is still an unsettled issue, we demonstrate by three examples that the accuracy achieved by the MQ solution cannot be rivaled by the FEM.