Volume 41, Issue 2
A Variational Analysis for the Moving Finite Element Method for Gradient Flows

J. Comp. Math., 41 (2023), pp. 191-210.

Published online: 2022-11

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• 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.

65M12, 65M55

xmxu@lsec.cc.ac.cn (Xianmin Xu)

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@Article{JCM-41-191, author = {Xu , Xianmin}, title = {A Variational Analysis for the Moving Finite Element Method for Gradient Flows}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {41}, number = {2}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2107-m2020-0227}, url = {http://global-sci.org/intro/article_detail/jcm/21176.html} }
TY - JOUR T1 - A Variational Analysis for the Moving Finite Element Method for Gradient Flows AU - Xu , Xianmin JO - Journal of Computational Mathematics VL - 2 SP - 191 EP - 210 PY - 2022 DA - 2022/11 SN - 41 DO - http://doi.org/10.4208/jcm.2107-m2020-0227 UR - https://global-sci.org/intro/article_detail/jcm/21176.html KW - Moving finite element method, Convergence analysis, Onsager principle. AB -

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.

Xianmin Xu. (2022). A Variational Analysis for the Moving Finite Element Method for Gradient Flows. Journal of Computational Mathematics. 41 (2). 191-210. doi:10.4208/jcm.2107-m2020-0227
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