A fast direct algorithm for implementing a high-order finite element method on rectangles as applied to boundary value problems for the Poisson equation

Fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for a high-order finite element method (FEM) are proposed based on the fast discrete Fourier transform. They generalize logarithmically optimal Fourier algorithms for solving boundary value problems for Poisson-type equations on rectangular meshes to high-order FEM. The algorithms can be extended to the multidimensional case and can be applied to nonstationary problems.