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.
A Trace--Logarithmic Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation
Xiangtan University（湘潭大学）；Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, Xiangtan University（湘潭大学基础学科基本算法理论与新型计算方法湖南省研究中心）；Hunan International Scientific and Technological Innovation Cooperation Base of Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University（湘潭大学计算科学湖南省国际科技合作基地、教育部智能计算与信息处理重点实验室）；National Center of Applied Mathematics in Hunan, Hunan Key Laboratory for Computation and Simulation in Science and Engineering（湖南应用数学中心、湖南省计算与仿真科学重点实验室）
