Energy stable and convergent finite element schemes for the modified phase field crystal equation^{∗,∗∗,∗∗∗}

¹ Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi 9, 20133 Milano, Italy.
maurizio.grasselli@polimi.it
² Laboratoire de Mathématiques et Applications UMR CNRS 7348, Université de Poitiers, Téléport 2 - BP 30179, boulevard Marie et Pierre Curie, 86962 Futuroscope Chasseneuil, France.
Morgan.Pierre@math.univ-poitiers.fr

Received: 24 February 2015
Revised: 19 October 2015
Accepted: 24 November 2015

Abstract

We propose a space semi-discrete and a fully discrete finite element scheme for the modified phase field crystal equation (MPFC). The space discretization is based on a splitting method and on a Galerkin approximation in H¹ for the phase function. This formulation includes the classical continuous finite elements. The time discretization is a second-order scheme which has been introduced by Gomez and Hughes for the Cahn–Hilliard equation. The fully discrete scheme is shown to be unconditionally energy stable and uniquely solvable for small time steps, with a smallness condition independent of the space step. Using energy estimates, we prove that in both cases, the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameters tend to 0. This is the first proof of convergence for the scheme of Gomez and Hughes, which has been shown to be unconditionally energy stable for several Cahn–Hilliard related equations. Using a Łojasiewicz inequality, we also establish that the discrete solution tends to a stationary solution as time goes to infinity. Numerical simulations with continuous piecewise linear (P¹) finite elements illustrate the theoretical results.