Numerical Analysis of Partial Differential Equations Using Maple and MATLAB

This is a brief introduction to numerical methods for elliptic partial differential equations (PDEs). It was designed to support a one-semester course for advanced undergraduates and beginning graduate students in mathematics, engineering and computational science. The authors’ primary goal is to describe the four main discretization techniques for developing numerical solutions of elliptic PDEs. Convergence proofs are provided for each method in settings that are simplified but illustrative of the major analytic techniques that are necessary for more detailed study. Codes are provided in MATLAB for each of the methods that give examples of basic implementations of each method.

Prerequisites for the book include multivariable calculus, basic linear algebra and preferably some experience with ordinary differential equations. The authors begin with an introduction to differential equations (ordinary and partial) aimed at readers with limited previous exposure to their study. In itself this is an attractive short course with historical background and examples of the canonical PDEs. The authors offer compact summaries not just for the usual suspects (the heat and wave equations) but also the reaction-diffusion, Maxwell and Navier-Stokes equations.

The four main discretization methods are treated roughly in order of increasing complexity. The methods are the finite difference, finite volume, spectral and finite element methods. The finite difference method approximates the derivatives in the PDE using a truncated Taylor series in each variable. This leads eventually to a system of linear equations with entries for each point on a grid. It is the simplest and most intuitive method, and is easiest to use for rectangular domains with Dirichlet-type boundary conditions. The finite volume method is designed to adapt to non-rectangular domains with various boundary conditions, but it adds a significant layer of complexity.

Spectral methods take an entirely different approach to solving PDEs, one that can give better approximations than finite difference or finite volume methods. Spectral methods look for a solution having the form of a truncated series of weighted basis functions. The most commonly used basis functions are the complex exponentials associated with discrete Fourier series. For non-periodic problems the authors suggest using Chebyshev polynomials, but these usually require using grid points that are non-equidistant.

Finite element methods are based on the calculus of variations. They use a systematic method of creating discretizations of PDEs and they work for arbitrary geometries and meshes. These methods are more complex than finite difference methods, but they provide the basis for many commercial codes that require solutions of PDEs.

Each section in the book has at least a few associated exercises. My favorite asks the students to apply the finite element method with the two-dimensional Helmholtz equation to simulate cooking a whole chicken in a microwave.

This text moves very quickly through many complex topics. The authors expect a good deal from the reader, particularly in the sections of the book that treat convergence questions. The book might best be used to make students aware of the alternative methods for numerical solution of PDEs, their basic features and limitations, and the convergence questions that are associated with them. By itself it will not make them skilled practitioners of the art. But the book does give students a sense of the many issues that are involved in commercially available software packages that must solve PDEs.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.