A VARIATIONAL ANALYSIS FOR THE MOVING FINITE ELEMENT METHOD FOR GRADIENTFLOWS  

作　　者：Xianmin Xu 

机构地区：[1]LSEC,Institute of Computational Mathematics and Scientific/Engineering Computing,NCMIS,AMSS,Chinese Academy of Sciences,Beijing 100190,China [2]School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

出　　处：《Journal of Computational Mathematics》2023年第2期191-210,共20页计算数学（英文）

基　　金：supported in part by NSFC grants DMS-11971469;the National Key R&D Program of China under Grant 2018YFB0704304 and Grant 2018YFB0704300.

摘　　要：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.

关 键 词：Moving finite element method Convergence analysis Onsager principle 

分 类 号：O241.82[理学—计算数学]