A Variational Analysis for the Moving Finite Element Method for Gradient Flows

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.