We study a system of semi-linear elliptic partial differential equations with a lower order cubic nonlinear term, and inhomogeneous Dirichlet boundary conditions, relevant for two-dimensional bistable liquid crystal devices, within a reduced Landau-de Gennes framework. The main results are (i) a priori error estimates for the energy norm and the $L^2$ norm, within the Nitsche's and discontinuous Galerkin frameworks under milder regularity assumptions on the exact solution and (ii) a reliable and efficient {\it a posteriori} analysis for a sufficiently large penalization parameter and a sufficiently fine triangulation in both cases. Numerical examples that validate the theoretical results, are presented separately.
翻译:我们研究一个在减少Landau-de Gennes框架内,与二维双层液晶晶装置相关的半线性椭圆性部分差分方程系统,其顺序较低,为非线性立方体,且不相容的dirichlet边界条件,主要结果如下:(一) 在关于确切解决办法的较温和的常规假设下,在Nitsche和不连续的Galerkin框架范围内,对能源规范及2美元标准进行先验误差估计;(二) 对足够大的惩罚性参数进行可靠和高效的后遗症分析,对这两种情况进行足够精细的三角测量;另外提出证实理论结果的数值实例。