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.
1. Introduction. 1.1 An illustrative example. 1.2 Transformation of problem. 1.3 Function spaces. 1.4 Variational reformulation. 1.5 A spectral method. 1.6 Numerical example. 1.7 Exterior problems. 2 Multivariate Polynomials. 2.1 Multivariate polynomials. 2.2 Triple recursion relation. 2.3 Rapid evaluation of orthonormal polynomials. 2.4 A Clenshaw algorithm. 2.5 Best approximation. 2.6 Quadrature over the unit disk, unit ball, and unit sphere. 2.7 Least squares approximation. 2.8 Matlab programs and numerical examples. 3 Creating Transformations of Regions. 3.1 Constructions of □ ф. 3.2 An integration-based mapping formula. 3.3 Iteration methods. 3.4 Mapping in three dimensions. 4 Galerkin's method for the Dirichlet and Neumann Problems. 4.1 Implementation. 4.2 Convergence analysis. 4.3 The Neumann problem. 4.4 Convergence analysis for the Neumann problem. 4.5 The Neumann problem with = 0. 4.6 De ning surface normals and Jacobian for a general surface. 5 Eigenvalue Problems. 5.1 Numerical solution - Dirichlet problem. 5.2 Numerical examples - Dirichlet problem. 5.3 Convergence analysis - Dirichlet problem. 5.4 Numerical solution - Neumann problem. 6 Parabolic problems. 6.1 Reformulation and numerical approximation. 6.2 Numerical examples. 6.3 Convergence analysis. 7 Nonlinear Equations. 7.2 Numerical examples. 7.3 Convergence analysis. 7.4 Neumann boundary value problem. 8 Nonlinear Neumann Boundary Value Problem. 8.1 The numerical method. 8.2 Numerical examples. 8.3 Error analysis. 8.4 An existence theorem for the three dimensional Stefan--Boltzmann problem. 9 The biharmonic equation. 9.1 The weak reformulation. 9.2 The numerical method. 9.3 Numerical Examples. 9.4 The eigenvalue problem. 10 Integral Equations. 10.1 Galerkin's numerical method. 10.2 Error analysis. 10.3 An integral equation of the rst kind