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Spectral methods are a class of spatial discretizations for differential equations. The key components for their formulation are the trial functions (also called the expansion or approximating functions) and the test functions (also known as weight functions). The trial functions, which are linear combinations of suitable trial basis functions, are used to provide the approximate representation of the solution. The test functions are used to ensure that the differential equation and perhaps some boundary conditions are satisfied as closely as possible by the truncated series expansion. This is achieved by minimizing, with respect to a suitable norm, the residual produced by using the truncated expansion instead of the exact solution. The residual accounts for the differential equation and sometimes the boundary conditions, either explicitly or implicitly. For this reason they may be viewed as a special case of the method of weighted residuals (Finlayson and Scriven (1966)). An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions. From this perspective, spectral methods may be viewed as a special case of Petrov-Galerkin methods (Zienkiewicz and Cheung (1967), Babuška and Aziz (1972)).

The expansion of a function u in terms of an infinite sequence of orthogonal functions {}, e.g., = Σ or = Σ, underlies many numerical methods of approximation. The accuracy of the approximations and the efficiency of their implementation influence decisively the domain of applicability of these methods in scientific computations.

For the remainder of this book we shall be concerned with the use of spectral methods to obtain approximate solutions to ordinary differential equations (ODEs) and, especially, partial differential equations (PDEs). With very few exceptions spectral methods have only been applied to the approximation of spatial, and not temporal, derivatives. Our focus in this book is on the spatial approximations. The reader is referred to Appendix D for a review of time-discretization methods, including some brief comments on spectral approximations to time derivatives. When all or part of the time discretization is implicit, then the solution of implicit equations is required to advance in time. This topic is covered in the following chapter.

The solution of implicit equations is an important component of many spectral algorithms. For steady problems this task is unavoidable, while spectral algorithms for many unsteady problems are only feasible if they incorporate implicit (or semi-implicit) time discretizations (see Appendix D for general information about time discretizations and Sect. 3.3 and CHQZ3, Chap. 3 for some uses of implicit time discretizations with spectral discretizations in space). We concentrate on linear systems, assuming that nonlinear ones are attacked by standard linearization techniques.

In the remainder of this book we concentrate on summarizing the fundamental spectral methods theory for approximation errors, stability and convergence, and apply this to the analysis of model equations. We will not present here all the details of all the proofs of the results that are cited. Rather, we illustrate the basic principles of the theory by presenting proofs for representative results. In many cases these proofs are delayed until later in the chapter (in the interests of having a coherent summary). For the same reason bibliographic references for the main contributions to the theory are likewise deferred to the end of the appropriate section.

In this chapter we present a fairly general approach to the stability and convergence analysis of spectral methods. We confine ourselves to linear problems. Analysis of several nonlinear problems is presented in Chap. 7 and in CHQZ3, Chap. 3. For time-dependent problems, only the discretizations in space are considered. Stability for fully discretized time-dependent problems is discussed in Appendix D by a classical eigenvalue analysis and in Chap. 7 by variational methods.

In this chapter, we apply the techniques for the theoretical analysis of spectral approximations to some differential operators and differential equations that are representative building blocks of the mathematical modelling in continuum mechanics. We first study the Poisson equation, followed by singularly perturbed elliptic equations that model advection-diffusion and reactiondiffusion processes featuring sharp boundary layers. Subsequently, we develop an eigenvalue analysis for several matrices produced by spectral approximations to diffusion, advection-diffusion and pure advection problems. We extend our analysis to the closely related study of the low-order preconditioning of spectral matrices.