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 - <p style="text-align: justify;">By using the Onsager principle as an approximation tool, we give a novel&nbsp; 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&nbsp; approximation theory of free-knot&nbsp; piecewise polynomials. We show that under certain&nbsp; 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&nbsp; for a&nbsp; linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.</p>