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Zbl 0906.65110
Guo, Ben-Yu
Spectral methods and their applications. (English)
[B] Singapore: World Scientific. viii, 349~p. \sterling~33.00/hbk (1998). ISBN 981-02-3333-7/hbk; ISBN 978-981-281-664-1/ebook

Spectral methods for the numerical solution of partial differential equations are introduced. This book presents the basic algorithms, the main theoretical results and some applications of spectral methods. In Chapter 1 a colloquial introduction is given. In Chapter 2 various orthogonal approximations in Sobolev spaces are discussed. Chapter 3 contains the theory of stability and convergence. In Chapter 4 applications to nonlinear problems are given. Furthermore spectral penalty methods, the spectral viscosity methods and spectral approximations to isolated solutions are considered. Chapter 5 is devoted to spectral approximations of multi-dimensional and high-order problems. Spectral domain decomposition and multigrid methods are also introduced. In the Chapters 6 and 7 spectral methods for semi-periodic problems and combined spectral methods are investigated. The final chapter focuses on spectral methods on the spherical surface.
[W.Heinrichs (Essen)]

MSC 2000:: *65N35 Collocation methods (BVP of PDE)
65-02 Research monographs (numerical analysis)
65N12 Stability and convergence of numerical methods (BVP of PDE)
35J25 Second order elliptic equations, boundary value problems
35J65 (Nonlinear) BVP for (non)linear elliptic equations

Keywords: spectral methods; algorithms; stability; convergence; Sobolev spaces; nonlinear problems; penalty methods; spectral viscosity methods; spectral domain decomposition; multigrid methods

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