A high-order spectral element fast Fourier transform for the Poisson equation

The aim of this work is to propose a novel, fast solver for the Poisson problem discretized with high-order spectral element methods (HO-SEM) in canonical geometries (rectangle in two dimensions, rectangular parallelepiped in three dimensions). This method is based on the use of the discrete Fourier transform to reduce the problem to the inversion of the symbol of the operator in the frequency space. The solver proposed is endowed with several properties. First, it preserves the efficiency of the standard FFT algorithm; then, the matrix storage is drastically reduced (in particular, it is independent of the space dimension); a pseudoexplicit singular value decomposition is used for the inversion of the symbols; and finally, it can be extended to nonperiodic boundary conditions. Furthermore, due to the underlying HO-SEM discretization, the multidimensional symbol of the operator can be efficiently computed from the one-dimensional symbol by tensorisation.