Numerical modeling of anisotropic ferroelectric materials with hybridizable discontinuous Galerkin methods

We investigate a gradient flow structure of the Ginzburg--Landau--Devonshire (GLD) model for anisotropic ferroelectric materials by reconstructing its energy form. We show that the modified energy form admits at least one minimizer. Under some regularity assumptions for the electric charge distribution and the initial polarization field, we prove that the $L^2$ gradient flow structure has a unique solution. To simulate the GLD model numerically, we propose an energy-stable semi-implicit time-stepping scheme and a hybridizable discontinuous Galerkin method for space discretization. Some numerical tests are provided to verify the stability and convergence of the proposed numerical scheme as well as some properties of ferroelectric materials.