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We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor products finite element method (FEM) based on the tensor products of the 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds for solving the $N$-dimensional Poisson type equation $-\Delta u+\alpha u=f$ ($N\geq 2$) with the Dirichlet boundary conditions. They are based on the well-known Fourier approaches. The key new points are a detailed description for the eigenpairs of the 1D eigenvalue problems for the high order FEM as well as the fast direct and inverse algorithms for expansion in the respective eigenvectors utilizing simultaneously several versions of the FFT (fast Fourier transform). Results of numerical experiments in 2D and 3D cases are presented.