We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which, we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.

翻译：本文研究了电场作用下液晶的Landau-de Gennes Q-张量模型，并发展了用于求解该模型的全离散数值格式。该格式采用体势能的凸分裂方法，并引入Q-张量截断算子以确保问题的适定性。我们证明了该格式的稳定性与适定性。最后，通过对格式中可容许参数施加限制，我们证明在时间步长和网格细化过程中，全离散格式的解（在子序列意义下）收敛于Q-张量模型的弱解。随后，我们展示了由该数值格式计算得到的数值结果，其中表明该格式能够模拟弗雷德里克斯转变过程。