Superconvergence of high order FEMs for eigenvalue problems with periodic boundary conditions

We study Adini’s elements for nonlinear Schrödinger equations (NLS) defined in a square box with periodic boundary conditions. First we transform the time-dependent NLS to a time-independent stationary state equation, which is a nonlinear eigenvalue problem (NEP). A predictor–corrector continuation method is exploited to trace solution curves of the NEP. We are concerned with energy levels and superfluid densities of the NLS. We analyze superconvergence of the Adini elements for the linear Schrödinger equation defined in the unit square. The optimal convergence rate O(h6)O(h6) is obtained for quasiuniform elements. For uniform rectangular elements, the superconvergence O(h6+p)O(h6+p) is obtained for the minimal eigenvalue, where p=1p=1 or p=2p=2. The theoretical analysis is confirmed by the numerical experiments. Other kinds of high order finite element methods (FEMs) and the superconvergence property are also investigated for the linear Schrödinger equation. Finally, the Adini elements-continuation method is exploited to compute energy levels and superfluid densities of a 2D Bose–Einstein condensates (BEC) in a periodic potential. Numerical results on the ground state as well as the first few excited-state solutions are reported.