A two-grid method is proposed to solve phase field models, arising from modeling the band gap optimization of photonic crystal design based on the Transverse Electric field (TE mode). In our approach, finite element methods on nested grids are used to solve different Partial Differential Equations (PDEs), say, the Hemholtz equation for the state constrain and the phase field equation for topology evolving, respectively. Dicretization on the coarse and fine grids are performed to balance the costs between the phase evolving and state equation solver. To fuse the two scheme together, additional correction step is necessary in two-grid settings. Furthermore, convergence analysis for the proposed two-grid scheme is established. Finally, numerical examples are presented to show the efficiency and robustness, and it is illustrated that the two-grid method improves the speed of computation while remain the computational accuracy.

提出了一种两网格方法来求解相场模型，这是基于对基于横向电场（TE模式）的光子晶体设计的带隙优化建模而产生的。在我们的方法中，使用嵌套网格上的有限元方法来求解不同的偏微分方程（PDE），例如分别用于状态约束的Hemholtz方程和用于拓扑演化的相场方程。对粗网格和细网格进行离散化，以平衡相位演化和状态方程求解器之间的成本。为了将两种方案融合在一起，在两网设置中需要额外的校正步骤。此外，针对所提出的两网格方案建立了收敛性分析。最后，通过数值示例说明了效率和鲁棒性，