Discontinuous Galerkin composite finite element methods (DGCFEM) are designed totackle approximation problems on complicated domains. Partial differential equationsposed 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 thispaper, an optical lattice will be used to illustrate the performance of the approximationalgorithm for the ground state computation of a Gross–Pitaevskii equation, which isan eigenvalue problem with eigenvector nonlinearity. We will adapt the convergenceresults of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penaltyhp-adaptive algorithm against the performance of the hp-DGCFEM.