Convergence Analysis and Numerical Implementation of a Second Order   Numerical Scheme for the Three-Dimensional Phase Field Crystal Equation
  In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in [46], with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation. 
Lixiu Dong, Wenqiang Feng, Cheng Wang, Steven M. Wise, Zhengru Zhang
math.NA