Year: 2023
Author: Xianmin Xu
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 2 : pp. 191–210
Abstract
By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2107-m2020-0227
Journal of Computational Mathematics, Vol. 41 (2023), Iss. 2 : pp. 191–210
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Moving finite element method Convergence analysis Onsager principle.