Mesh sensitivity for numerical solutions of phase-field equa- tions using r-adaptive finite element methods

There have been several recent works on developing moving mesh methods for solving phase-field equations. However, it is observed that some of these moving mesh solutions are essentially different with the solutions on very fine fixed meshes. One of the purposes of this work is to understand the reason for the differences. We carried out numerical sensitivity studies systematically in this paper and it can be concluded that for the phase-field equations, the numerical solutions are very sensitive to the starting mesh and the monitor function. At the same time, an efficient alternating Crank-Nicolson time discretization scheme is developed for solving the nonlinear system resulting from a finite element approximation to the phase-field equations. AMS subject classfications: 65M20, 65M50, 65M60, 80A22