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**Spectral methods. Fundamentals in single domains.**
*(English)*
Zbl 1093.76002

Scientific Computation. Berlin: Springer (ISBN 3-540-30725-7/hbk). xxiii, 563 p. (2006).

Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between theoretical and practical aspects of spectral methods that was the hallmark of the earlier book [the first edition: C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral methods in fluid dynamics. New York etc.: Springer-Verlag (1988; Zbl 0658.76001); 2nd corr. printing: C. Canuto et al., Spectral methods in fluid dynamics. Berlin etc.: Springer-Verlag (1991; Zbl 0717.76004)], now the authors include many improvements in algorithms and in the theory of spectral methods that have been made since 1991. The main aim of the book is to discuss the approximations of solutions to ordinary and partial differential equations in single domains by expansions in smooth, global basis functions. The first half of the book provides algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods in single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary value problems. Both the algorithmic and theoretical discussions cover spectral methods in tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on Galerkin method with numerical integration in spectral methods. The discussion of direct and iterative solution methods is endowed with numerical examples that illustrate the key properties of various spectral approximations and solution algorithms. According to authors, a companion book [C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral methods. Evolution to complex geometries and applications to fluid dynamics. Berlin: Springer (2007; Zbl 1121.76001)] contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods in complex geometries and provides detailed discussions of fluid dynamics applications in simple and complex geometries.

Reviewer: Nina Shokina (Stuttgart)