Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations

ii Preface Radial Basis Function (RBF) methods have become the primary tool for interpolating multidimensional scattered data. RBF methods also have become important tools for solving Partial Differential Equations (PDEs) in complexly shaped domains. Classical methods for the numerical solution of PDEs (finite difference, finite element, finite volume, and pseudospectral methods) are based on polynomial interpolation. Local polynomial based methods (finite difference, finite element, and finite volume) are limited by their algebraic convergence rates. Numerical studies, such as the comparison of the MQ collocation method with the finite element method in [136], have been done that illustrate the superior accuracy of the MQ method when compared to local polynomial methods. Global polynomial methods, such as spectral methods, have exponential convergence rates but are limited by being tied to a fixed grid. RBF methods are not tied to a grid and in turn belong to a category of methods called meshless methods. The large number of recent books, on meshfree methods illustrates the popularity that the methods have recently enjoyed. The global, non-polynomial, RBF methods may be successfully applied to achieve exponential accuracy where traditional methods either have difficulties or fail. An example is in multidimensional problems in non-rectangular domains. RBF methods succeed in very general settings by composing a univariate function with the Euclidean norm which turns a multidimensional problem into one that is virtually one dimensional. RBF methods are a generalization of the Multiquadric (MQ) RBF method which utilizes one particular RBF. The MQ RBF method has a rich history of theoretical development and applications. The subject of this monograph is the MQ RBF approximation method with a particular emphasis on using the method to numerically solve partial differential equations. This monograph differs from other recent books [31, 63, 179, 209] on meshless methods in that it focuses only on the MQ RBF while others have focused on meshless methods in general. It is hoped that this refined focus will result in a clear and concise exposition of the area. Matlab code that illustrates key ideas about the implementation of the MQ method has been included in the text of the manuscript. The included code, as well as additional Matlab code used to produce many of the numerical examples, can be found on the web at iii A Matlab programs 147 B Additional Meshfree Method References 153 viii CONTENTS