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.

翻译：本文通过重构各向异性铁电材料Ginzburg-Landau-Devonshire (GLD)模型的能量形式，研究其梯度流结构。研究表明该修正能量形式至少存在一个极小值。在电荷分布与初始极化场满足若干正则性假设的条件下，我们证明了该$L^2$梯度流结构存在唯一解。为数值模拟GLD模型，我们提出了能量稳定的半隐式时间推进格式，并采用混合不连续伽辽金方法进行空间离散。通过数值实验验证了所提数值格式的稳定性、收敛性以及铁电材料的若干特性。