research-article

Matrix-oriented FEM formulation for reaction-diffusion PDEs on a large class of 2D domains

Authors:

Massimo Frittelli,

Ivonne SguraAuthors Info & Claims

Volume 200, Issue C

Pages 286 - 308

https://doi.org/10.1016/j.apnum.2023.07.010

Published: 09 July 2024 Publication History

Abstract

For the spatial discretization of elliptic and parabolic partial differential equations (PDEs), we provide a Matrix-Oriented formulation of the classical Finite Element Method, called MO-FEM, of arbitrary order k ∈ N. On a quite general class of 2D domains, namely separable domains, and even on special surfaces, the discrete problem is then reformulated as a multiterm Sylvester matrix equation where the additional terms account for the geometric contribution of the domain shape. By considering the IMEX Euler method for the PDE time discretization, we obtain a sequence of these equations. To solve each multiterm Sylvester equation, we apply the matrix-oriented form of the Preconditioned Conjugate Gradient (MO-PCG) method with a matrix-oriented preconditioner that captures the spectral properties of the Sylvester operator. Solving the Poisson problem and the heat equation on some separable domains by MO-FEM-PCG, we show a gain in computational time and memory occupation wrt the classical vector PCG with same preconditioning or wrt a LU based direct method. As an application, we show the advantages of the MO-FEM-PCG to approximate Turing patterns on some separable domains and cylindrical surfaces for a morphochemical reaction-diffusion PDE system for battery modelling.

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cover image Applied Numerical Mathematics

Applied Numerical Mathematics Volume 200, Issue C

Jun 2024

466 pages

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Elsevier Science Publishers B. V.

Netherlands

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Published: 09 July 2024

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