Energy stable and convergent finite element schemes for the modified phase field crystal equation (Q2830707)

The authors make a further progress in their line of exploration of the mathematical properties of the modified phase field crystal equation (MPFC), namely, the phase field crystal equation (PFC), but with an additional relaxation term (second-order in time) previously proposed to account for elastic interactions [\textit{P. Stefanovic}, \textit{M. Haataja} and \textit{N. Provatas}, ``Phase-field crystals with elastic interactions'', Phys. Rev. Lett. 96, No. 22, Article ID 225504, 4 p. (2006; \url{doi:10.1103/PhysRevLett.96.225504})] or for large deviations from thermodynamic equilibrium [\textit{P. Galenko} and \textit{D. Jou}, ``Diffuse-interface model for rapid phase transformations in nonequilibrium systems'', Phys. Rev. E 71, No. 4, Article ID 046125, 13 p. (2005; \url{doi:10.1103/PhysRevE.71.046125})]. In contrast to the PFC equation, which is a gradient flow of a non-local free energy depending on the phase function \(u\), the MPFC requires to add to such free energy a kinetic term depending on the time derivative of \(u\). In recent previous papers, for instance [\textit{M. Grasselli} and \textit{H. Wu}, Math. Models Methods Appl. Sci. 24, No. 14, 2743--2783 (2014; Zbl 1304.35690)], one of the authors studied properties of the MPFC convergence properties of the MPFC with a continuous space. Here, instead, the authors generalize the analysis to a space-discretized version of the equation. They propose a semi-discrete and fully discrete finite element scheme, based on a splitting method and on a Galerkin approximation in \(H^1\) for the phase function. The time discretization follows a second-order scheme introduced for the Cahn-Hilliard equation [\textit{H. Gomez} and \textit{T. J. R. Hughes}, J. Comput. Phys. 230, No. 13, 5310--5327 (2011; Zbl 1419.76439)]. The main results of the paper are the following ones. NEWLINENEWLINEA) 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. NEWLINENEWLINEB) Using energy estimates, and assuming only some natural conditions on the initial conditions, it is shown that in both cases the discrete solution converges to the unique energy solution of the MPFC equation as the discretization parameter tends to 0. This is claimed to be the first proof of convergence for the scheme of Gomez and Hughes [loc. cit.], which has been shown to be unconditionally stable for several Cahn-Hilliard related equations. NEWLINENEWLINEC) It is established that the discrete solution tends to a stationary solution as time goes to infinity, a result which is far from being trivial because the set of steady states can be very complicated. NEWLINENEWLINED) Finally, numerical results in one and two space dimensions with continuous piecewise linear (P1) finite elements explicitly illustrate some aspects of the theoretical results.