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. Features -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.