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Spectral Methods Using Multivariate Polynomials on the Unit Ball

Author: Kendall E Atkinson; David Chien; Olaf Hansen
Publisher: Milton : CRC Press LLC, 2019.
Series: Chapman and Hall/CRC Monographs and Research Notes in Mathematics Ser.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
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  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Atkinson, Kendall
Spectral Methods Using Multivariate Polynomials on the Unit Ball
Milton : CRC Press LLC,c2019
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Kendall E Atkinson; David Chien; Olaf Hansen
ISBN: 9781000725865 1000725863 9781000725988 1000725987 9781000725926 1000725928 9780429344374 0429344376
OCLC Number: 1129224261
Notes: Description based upon print version of record.
8.4 An existence theorem for the three-dimensional Stefan-Boltzmann problem
Description: 1 online resource (275 p.).
Contents: 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
Series Title: Chapman and Hall/CRC Monographs and Research Notes in Mathematics Ser.

Abstract:

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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