A Fast Direct Algorithm for Implementing a High-Order Finite Element Method on Rectangles as Applied to Boundary Value Problems for the Poisson Equation
A new fast direct algorithm for implementing a finite element method (FEM) of order on rectangles as applied to boundary value problems for Poisson-type equations is described that extends a well-known algorithm for the case of difference schemes or bilinear finite elements (n = 1). Its core consists of fast direct and inverse algorithms for expansion in terms of eigenvectors of one-dimensional eigenvalue problems for an nth-order FEM based on the fast discrete Fourier transform. The amount of arithmetic operations is logarithmically optimal in the theory and is rather attractive in practice. The algorithm admits numerous further applications (including the multidimensional case).