On some applications of the superposition principle with Fourier basis

This paper presents several applications of (local) Fourier basis combined with corrector techniques via the superposition principle to compute solutions of boundary value problems. Our methodology is inspired by the well-known corrector technique used in asymptotic singular perturbation theory-see, for example, [W. Eckhaus, Asymptotic Analysis of Singular Perturbations, North-Holland, Amsterdam, 1979]. We build solvers for time dependent boundary value problems and/or singular perturbation problems using Fourier expansions combined to additional convenient set of time independent basis functions to enforce the boundary conditions. The algorithms are very well suited for parallel computing because they rely mainly on FFTs and basis functions given analytically or computed (in parallel) once and for all. Our method can be spectral accurate for simple geometry. We obtain in addition interesting new results for boundary value problems with complex geometry. We describe in this paper the implementation of the method and the numerical results for many linear and nonlinear problems: our goal is to test the advantages and the limits of our approach in order to motivate further theoretical investigations. Our method applied to steady one-dimensional problems is similar to the work of Israeli, Vozovoi, and Averbuch [J. Sci. Comput., 8 (1993), pp. 135-149] on domain decomposition with local Fourier basis for the Helmotz problem. However our methodology for time-dependent problem and/or two space dimensions geometry is rather different in its spirit and complements previous investigations of Israeli, Vozovoi, and Averbuch.