PREFACEIn the past two decades, spectral methods have developed rapidly. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to numerical simulations in various fields.The purpose of this book is to present the basic algorithms, the main theoretical results and some applications of spectral methods. Particular attention is paid to the applications to nonlinear problems.The outline of this book is as follows. Chapter 1 is a colloquial introduction to spectral methods. In Chapter 2, we discuss various orthogonal approximations in Sobolev spaces, used in spectral methods. We also discuss the filterings and recov ering the spectral accuracy. Chapter 3 is a survey of the theory of stability and convergence. Chapter 4 consists of two parts. In the first part, we present some basic spectral methods with their applications to nonlinear problems. In the second part, we consider the spectral penalty methods, the spectral viscosity methods and the spectral approximations of isolated solutions. Chapter 5 is devoted to the spectral approximations of multi-dimensional and high order problems. The spectral domain decomposition methods and the spectral multigrid methods are also introduced. We consider the mixed spectral methods for semi-periodic problems in Chapter 6, and some combined spectral methods with their applications in Chapter 7. The final chapter focuses on the spectral methods on the spherical surface.
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