  Electric field plays an important role in ferroelectric phase transition.
There have been numerous phase field formulations attempting to account for
electrostatic interactions subject to different boundary conditions. In this
paper, we develop new variational forms of the phase field electrostatic energy
and the relaxation dynamics of the polarization vector that involves a hybrid
representation in both real and Fourier variables. The new formulations avoid
ambiguities appeared in earlier studies and lead to much more effective ways to
perform variational studies and numerical simulations. Computations of
polarization switching in a single domain by applying the new formulations are
provided as illustrative examples.
