  Phase field methods have been widely used to study phase transitions and
polarization switching in ferroelectric thin films. In this paper, we develop
an efficient numerical scheme for the variational phase field model based on
variational forms of the electrostatic energy and the relaxation dynamics of
the polarization vector. The spatial discretization combines the Fourier
spectral method with the finite difference method to handle three-dimensional
mixed boundary conditions. It allows for an efficient semi-implicit
discretization for the time integration of the relaxation dynamics. This method
avoids explicitly solving the electrostatic equilibrium equation (a Poisson
equation) and eliminates the use of associated Lagrange multipliers. We present
several numerical examples including phase transitions and polarization
switching processes to demonstrate the effectiveness of the proposed method.
