We consider a system of second-order nonlinear elliptic partial differential equations that models the equilibrium configurations of a two-dimensional planar bistable nematic liquid crystal device. Discontinuous Galerkin (dG) finite element methods are used to approximate the solutions of this nonlinear problem with nonhomogeneous Dirichlet boundary conditions. A discrete inf–sup condition demonstrates the stability of the dG discretization of a well-posed linear problem. We then establish the existence and local uniqueness of the discrete solution of the nonlinear problem. A priori error estimates in the energy and |$\mathbf{L}^2$| norms are derived and a best approximation property is demonstrated. Further, we prove the quadratic convergence of the Newton iterates along with complementary numerical experiments.

中文翻译：

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求解Landau-de Gennes液晶最小化问题的不连续Galerkin有限元方法

我们考虑一个二阶非线性椭圆偏微分方程系统，该系统对二维平面双稳态向列液晶器件的平衡构型进行建模。使用不连续的Galerkin（dG）有限元方法来逼近具有非齐次Dirichlet边界条件的该非线性问题的解决方案。离散的inf-sup条件证明了适当摆正的线性问题的dG离散化的稳定性。然后，我们建立了非线性问题离散解的存在性和局部唯一性。能量和| $ \ mathbf {L} ^ 2 $ |中的先验误差估计导出了范数，并证明了最佳逼近性质。此外，我们证明了牛顿迭代的二次收敛性以及互补的数值实验。