Stability and convergence of a second-order mixed finite element method for the Cahn-Hilliard equation

In this paper, we devise and analyse 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-infinity (0,T;L-infinity) and the discrete chemical potential is bounded in L-infinity (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.