Approximation of periodic solutions of two-mode phase-field crystal model

In present paper we consider a mathematical model of two-mode phase-field crystal (PFC). This model describes the microscopic evolution and ordering of matter during crystallization from the homogeneous phase. The model is represented by a nonlinear partial differential equation of the tenth order in space and second order in time. The solution of PFC-model was performed using the Galerkin finite-element method. Due to the periodic form of the numerical solutions of this model, the additional spatial scale appeared and so this requires an increased discretization accuracy. The mesh convergence criteria and discretization parameters for the numerical solutions is considered, taking into account the computational complexity of two-mode PFC-model. The influence of size of finite elements (FE) and their order of base functions on the approximation of the solution in FE is considered. The correspondent numerical solution is devoted to the motion of planar crystallization front. The optimal sizes of FEs are determined, and the efficiency of numerical simulations using various software packages and solvers is compared.