Spectral Methods and a Maximum Principle

Various spectral Chebyshev approximations of a model boundary layer problem for both a Helmholtz and an advection-diffusion operator are considered. It is assumed that simultaneously the boundary layer width tends to zero and the resolution power of the numerical method tends to infinity. The behavior of the spectral solutions in the frequency space and in the physical space is investigated. Error estimates are derived. 0. Introduction. Spectral methods using expansions in eigenfunctions of singular Sturm-Liouville operators (such as Chebyshev or Legendre polynomials) have been proven successful in the numerical approximation of various boundary value problems (see, e.g., [3], [8], [14] and the references therein). Among the features of these methods is the possibility of accurately representing boundary layers. Such a property is related to the high resolution power of the spectral basis functions near the boundary points, since it is there where they concentrate most of their extrema (see [8, Section 3]). Using a suitable coordinate transformation it is even possible to achieve infinite-order accuracy near the boundaries [13]. In this paper we analyze the behavior of various spectral approximations to the following model boundary value problems: eUxx + 5?U = 0, -Kx 0 is a constant and either £?U = U (as a model for a Helmholtz problem) or SfU — Ux (as a model for an advection-diffusion problem). The spectral solution u is a global polynomial of degree TV, expanded in terms of Chebyshev polynomials and defined by any of the discrete procedures popularly used in spectral methods, namely a Galerkin or a tau or a collocation scheme. We shall establish several properties of u which hold uniformly with respect to e and TV, and we shall derive estimates on the error u — U as e —► 0 and TV —> oo simultaneously. The exact solution U is uniformly bounded between 0 and 1 by the classical maximum principle (for complete studies on singular perturbation problems we refer, e.g., to [7], [10]). In contrast, the spectral solution is not positive throughout the domain if e is so small compared to TV-1 that a Gibbs phenomenon occurs at x = 1. In other words, the maximum principle in the physical space does not Received December 16, 1985; revised August 24, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 65N30; Secondary 65N35. *Part of this work was done while the author was visiting the School of Mathematics and the Institute for Mathematics and its Applications at the University of Minnesota in the Winter of 1985. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page