A direct fast FFT-based implementation for high order finite element method on rectangular parallelepipeds for PDE

TL;DR

This paper introduces a new FFT-based algorithm that efficiently implements high order finite element methods on rectangular parallelepipeds, significantly improving computational speed for solving Poisson and related PDEs.

Contribution

It presents a logarithmically optimal and practical direct FFT-based algorithm for high order FEM on multi-dimensional rectangles, enabling faster PDE solutions.

Findings

Algorithm is logarithmically optimal in theory

Implementation is fast in practice

Applicable to various time-dependent PDEs

Abstract

We present a new direct logarithmically optimal in theory and fast in practice algorithm to implement the high order finite element method on multi-dimensional rectangular parallelepipeds for solving PDEs of the Poisson kind. The key points are the fast direct and inverse FFT-based algorithms for decomposition in eigenvectors of the 1D eigenvalue problems for the high order FEM. The algorithm can further be used for numerous applications, in particular, to implement the high order finite element methods for various time-dependent PDEs.