Spectral Methods Using Multivariate Polynomials on the Unit Ball is a research level text on a numerical method for the solution of partial differential equations. The authors introduce, illustrate with examples, and analyze 'spectral methods' that are based on multivariate polynomial approximations. The method presented is an alternative to finite element and difference methods for regions that are diffeomorphic to the unit disk, in two dimensions, and the unit ball, in three dimensions. The speed of convergence of spectral methods is usually much higher than that of finite element or finite difference methods.
- Introduces the use of multivariate polynomials for the construction and analysis of spectral methods for linear and nonlinear boundary value problems
- Suitable for researchers and students in numerical analysis of PDEs, along with anyone interested in applying this method to a particular physical problem
- One of the few texts to address this area using multivariate orthogonal polynomials, rather than tensor products of univariate polynomials.
Kendall Atkinson is Professor Emeritus at University of Iowa as well as Fellow of the Society for Industrial & Applied Mathematics (SIAM). He received his PhD from University of Wisconsin - Madison and has had Faculty appointments at Indiana University, University of Iowa as well as Visiting appointments at Colorado State University, Australian National University, University of New South Wales, University of Queensland. His research interests include numerical analysis, integral equations, multivariate approximation, spectral methods
David Chien, PHD, is Professor in the Department of Mathematics at California State University San Marcos. He has authored journal articles in his areas of research interest, which include the numerical solution of integral equations and boundary integral equation methods.
Olaf Hansen is Professor of Mathematics, California State University San Marcos. He received his PhD from Johannes Gutenberg University, Mainz, Germany in 1994 and his research interests include Analysis and Numerical Approximation of Boundary and Initial Value Problems and Integral Equations.
Chapter 1: Introduction
Chapter 2: Multivariate Polynomials
Chapter 3: Creating Transformations of Regions
Chapter 4: Galerkin's method for the Dirichlet and Neumann Problems
Chapter 5: Eigenvalue Problems
Chapter 6: Parabolic problems
Chapter 7: Nonlinear Equations
Chapter 8: Nonlinear Neumann Boundary Value Problem
Chapter 9: The biharmonic equation
Chapter 10: Integral Equations