This book presents the basic algorithms, the main theoretical results, and some applications of spectral methods. Particular attention is paid to the applications of spectral methods to nonlinear problems arising in fluid dynamics, quantum mechanics, weather prediction, heat conduction and other fields.The book consists of three parts. The first part deals with orthogonal approximations in Sobolev spaces and the stability and convergence of approximations for nonlinear problems, as the mathematical foundation of spectral methods. In the second part, various spectral methods are described, with some applications. It includes Fourier spectral method, Legendre spectral method, Chebyshev spectral method, spectral penalty method, spectral vanishing viscosity method, spectral approximation of isolated solutions, multi-dimensional spectral method, spectral method for high-order equations, spectral-domain decomposition method and spectral multigrid method. The third part is devoted to some recent developments of spectral methods, such as mixed spectral methods, combined spectral methods and spectral methods on the surface.Contents:Orthogonal Approximations in Sobolev SpacesStability and ConvergenceSpectral Methods and Pseudospectral MethodsSpectral Methods for Multi-Dimensional and High Order ProblemsMixed Spectral MethodsCombined Spectral MethodsSpectral Methods on the Spherical SurfaceReadership: Mathematicians.Key Features:Offers state-of-the-art numerical and analytical approaches for predicting the response of buckling and postbuckling structuresPresents an extensive collection of test data on flat, curved, cylindrical and spherical structures — the result of comprehensive experimental test program conducted over many yearsProvides a number of strategies for the optimum design of lightweight structures