Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of the approximation algorithm for the ground state computation of a Gross–Pitaevskii equation, which is an eigenvalue problem with eigenvector nonlinearity. We will adapt the convergence results of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penalty hp-adaptive algorithm against the performance of the hp-DGCFEM.

不连续伽辽金复合有限元方法 (DGCFEM) 旨在解决复杂域上的逼近问题。当存在需要与宏观现象同时建模的细观或局部现象时，在复杂域上提出的偏微分方程很常见。在本文中，将使用光学晶格来说明用于 Gross-Pitaevskii 方程基态计算的近似算法的性能，这是一个具有特征向量非线性的特征值问题。我们将 Marcati 和 Maday 2018 的收敛结果适应这一特定类别的不连续逼近空间，并将经典对称内部惩罚 hp 自适应算法的性能与 hp-DGCFEM 的性能进行对比。