%0 Journal Article
%T Stability and Convergence of a Second Order Mixed Finite Element Method for the Cahn-Hilliard Equation
%A Amanda E. Diegel
%A Cheng Wang
%A Steven M. Wise
%J Mathematics
%D 2014
%I arXiv
%X In this paper we devise and analyze an unconditionally stable, second-order-in-time numerical scheme for the Cahn-Hilliard equation in two and three space dimensions. We prove that our two-step scheme is unconditionally energy stable and unconditionally uniquely solvable. Furthermore, we show that the discrete phase variable is bounded in $L^\infty (0,T;L^\infty)$ and the discrete chemical potential is bounded in $L^\infty (0,T;L^2)$, for any time and space step sizes, in two and three dimensions, and for any finite final time $T$. We subsequently prove that these variables converge with optimal rates in the appropriate energy norms in both two and three dimensions. We include in this work a detailed analysis of the initialization of the two-step scheme.
%U http://arxiv.org/abs/1411.5248v1