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We present direct logarithmically optimal in theory and fast in practicealgorithms to implement the tensor product high order finite element method onmulti-dimensional rectangular parallelepipeds for solving PDEs of the Poissonkind. They are based on the well-known Fourier approaches. The key new pointsare the fast direct and inverse FFT-based algorithms for expansion ineigenvectors of the 1D eigenvalue problems for the high order FEM. Thealgorithms can further be used for numerous applications, in particular, toimplement the tensor product high order finite element methods for varioustime-dependent PDEs. Results of numerical experiments in 2D and 3D cases arepresented.

On fast Fourier solvers for the tensor product high-order FEM for a generalized Poisson equation