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Phase field methods have been widely used to study phase transitions andpolarization switching in ferroelectric thin films. In this paper, we developan efficient numerical scheme for the variational phase field model based onvariational forms of the electrostatic energy and the relaxation dynamics ofthe polarization vector. The spatial discretization combines the Fourierspectral method with the finite difference method to handle three-dimensionalmixed boundary conditions. It allows for an efficient semi-implicitdiscretization for the time integration of the relaxation dynamics. This methodavoids explicitly solving the electrostatic equilibrium equation (a Poissonequation) and eliminates the use of associated Lagrange multipliers. We presentseveral numerical examples including phase transitions and polarizationswitching processes to demonstrate the effectiveness of the proposed method.

Numerical Discretization of Variational Phase Field Model for Phase Transitions in Ferroelectric Thin Films