A Modified Fourier–Galerkin Method for the Poisson and Helmholtz Equations

In this paper we present a modified Fourier–Galerkin method for the numerical solution of the Poisson and Helmholtz equations in a d-dimensional box. The inversion of the differential operators requires O(N ^d ) operations, where N ^d is the number of unknowns. The total cost of the presented algorithms is O(N ^d :log₂:N), due to the application of the Fast Fourier Transform (FFT) at the preprocessing stage. The method is based on an extension of the Fourier spaces by adding appropriate functions. Utilizing suitable bilinear forms, approximate projections onto these extended spaces give rapidly converging and highly accurate series expansions.