On approximation of a periodic solution of the phase field crystal equation in simulations by the finite elements method

The paper presents a mathematical model of the phase field crystal (PFC), describing the evolution of the microstructure of matter during the crystallization process. Such a model is expressed by a nonlinear particle differential equation of the sixth order in space and the second in time, for the solution of which in recent years finite element computational algorithms have been developed and guarantee unconditional stability and second order of convergence. However, due to the periodic nature of the solution of the PFC problem, the accuracy of the approximation of a numerical solution can vary significantly with a change in the discretization parameters of the simulated system.
Taking into account the high computational complexity of the PFC problem in the three-dimensional formulation, the determination of the discretization criteria becomes an urgent practical issue. In this article, we study the influence of finite element sizes on the approximation of the solution of the PFC problem for cases of a flat and spherical crystallization front. It is shown that the excess of certain dimensions of the final element leads to significant qualitative changes in the numerical solution and, as a consequence, to a sharp decrease in the accuracy of the approximation. (In Russian).