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

2017/01/14 by Alexander Zlotnik, Zlotnik, Alexander, Ilya Zlotnik +1
Engineering · Physics and Astronomy · #65F05 #65F15 #65M60 #65T99 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1701.03967

openalex publication_date 2017/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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