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

## TL;DR

This paper introduces efficient Fourier-based algorithms for high-order finite element methods on rectangular domains, enabling fast solutions to Poisson and related PDEs with optimal theoretical and practical performance.

## Contribution

It develops novel FFT-based algorithms for eigenvector expansions in high-order FEM, improving computational efficiency for multi-dimensional PDEs.

## Key findings

- Algorithms are logarithmically optimal in theory.
- Numerical experiments confirm practical efficiency in 2D and 3D.
- Applicable to various time-dependent PDEs.

## Abstract

We present direct logarithmically optimal in theory and fast in practice algorithms to implement the tensor product high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. They are based on the well-known Fourier approaches. The key new points are the fast direct and inverse FFT-based algorithms for expansion in eigenvectors of the 1D eigenvalue problems for the high order FEM. The algorithms can further be used for numerous applications, in particular, to implement the tensor product high order finite element methods for various time-dependent PDEs. Results of numerical experiments in 2D and 3D cases are presented.