Preconditioning for a Class of Spectral Differentiation Matrices
Journal of Scientific Computing
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In this paper, we study systems of functional equations (FEs) and first-order partial differential equations (PDEs) suggested in [SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607--647 and SIAM J. Numer. Anal., 29 (1992), pp. 1741--1768] as approximations for the computation of invariant tori. The main new ideas of this paper are, first, to investigate these systems in the setting of Hilbert spaces rather than in the setting of Banach spaces and, then, to employ Fourier methods instead of difference methods for a numerical solution. Based on the setting of Sobolev spaces Hs(Tp), proper conditions for the PDE and FE systems to be dissipative are described, and some regularity results for the FE system are proved. We studied two kinds of Fourier methods, the spectral method and the pseudospectral method, in detail under dissipativity conditions. Convergence and optimal error estimates are shown theoretically for these Fourier methods in the case of general linear systems. Numerical results for three examples provided in the last section indicate that the Fourier method behaves very well not only for smooth solutions but also for nonsmooth solutions.