Spectral Methods Using Multivariate Polynomials on the Unit Ball [electronic resource].

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Cover; Half Title; Series Page; Title Page; Copyright Page; Dedication; Contents; Preface; 1. Introduction; 1.1 An illustrative example; 1.2 Transformation of the problem; 1.3 Function spaces; 1.4 Variational reformulation; 1.5 A spectral method; 1.6 A numerical example; 1.7 Exterior problems; 1.7.1 Exterior problems in R3; 2. Multivariate Polynomials; 2.1 Multivariate polynomials; 2.2 Triple recursion relation; 2.3 Rapid evaluation of orthonormal polynomials; 2.3.1 Evaluating derivatives for the planar case; 2.3.2 Evaluating derivatives for the three-dimensional case

5. Eigenvalue Problems5.1 Numerical solution -- Dirichlet problem; 5.2 Numerical examples -- Dirichlet problem; 5.3 Convergence analysis -- Dirichlet problem; 5.4 Numerical solution -- Neumann problem; 5.4.1 Numerical examples -- Neumann problem; 6. Parabolic Problems; 6.1 Reformulation and numerical approximation; 6.1.1 Implementation; 6.2 Numerical examples; 6.2.1 An example in three dimensions; 6.3 Convergence analysis; 6.3.1 Further comments; 7. Nonlinear Equations; 7.1 A spectral method for the nonlinear Dirichlet problem; 7.2 Numerical examples; 7.2.1 A three-dimensional example

7.3 Convergence analysis7.3.1 A nonhomogeneous boundary condition; 7.4 Neumann boundary value problem; 7.4.1 Implementation; 7.4.2 Numerical example; 7.4.3 Handling a nonzero Neumann condition; 8. Nonlinear Neumann Boundary Value Problems; 8.1 The numerical method; 8.1.1 Solving the nonlinear system; 8.2 Numerical examples; 8.2.1 Another planar example; 8.2.2 Two three-dimensional examples; 8.3 Error analysis; 8.3.1 The linear Neumann problem; 8.3.2 The nonlinear Neumann problem; 8.3.3 The error over; 8.3.4 A nonhomogeneous boundary value problem

8.4 An existence theorem for the three-dimensional Stefan-Boltzmann problem

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.

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