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Fast Fourier Solvers for the Tensor Product High-Order FEM for a Poisson Type Equation

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Abstract

Logarithmically optimal in theory and fast in practice, direct algorithms for implementing a tensor product finite element method (FEM) based on tensor products of 1D high-order FEM spaces on multi-dimensional rectangular parallelepipeds are proposed for solving the N-dimensional Poisson-type equation \( - \Delta u + \alpha u = f\) (\(N \geqslant 2\)) with Dirichlet boundary conditions. The algorithms are based on well-known Fourier approaches. The key new points are a detailed description of the eigenpairs of the 1D eigenvalue problems for the high-order FEM, as well as fast direct and inverse eigenvector expansion algorithms that simultaneously employ several versions of the fast Fourier transform. Results of numerical experiments in the 2D and 3D cases are presented. The algorithms can be used in numerous applications, in particular, to implement tensor product high-order finite element methods for various time-dependent partial differential equations, including the multidimensional heat, wave, and Schrödinger ones.

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Funding

The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics in 2019–2020 (grant no. 19-01-021) and was supported by the Russian Academic Excellence Project “5-100” and the Russian Foundation for Basic Research, grant no. 19-01-00262.

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Authors and Affiliations

  1. National Research University Higher School of Economics, 109028, Moscow, Russia

    A. A. Zlotnik

  2. Settlement Depository Company, 115419, Moscow, Russia

    I. A. Zlotnik

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  1. A. A. Zlotnik
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  2. I. A. Zlotnik
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Correspondence to A. A. Zlotnik or I. A. Zlotnik.

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Zlotnik, A.A., Zlotnik, I.A. Fast Fourier Solvers for the Tensor Product High-Order FEM for a Poisson Type Equation. Comput. Math. and Math. Phys. 60, 240–257 (2020). https://doi.org/10.1134/S096554252002013X

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  • DOI: https://doi.org/10.1134/S096554252002013X

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