Superconvergence of bi-k-Lagrange elements for eigenvalue problems

We study superconvergence of bi-k-Lagrange elements for parameter-dependent problems where k⩾2. We show that the superconvergence rate of the bi-k-Lagrange elements is two orders higher than that of the kth-order Lagrange elements. This is a significant improvement of the previous results [C.-S. Chien, H.T. Huang, B.-W. Jeng, Z.C. Li, Superconvergence of FEMs and numerical continuation for parameter-dependent problems with folds, Int. J. Bifurcation Chaos 18 (2008) 1321-1336], which is only one order (or a half order) higher than that of the latter. Next, we apply the bi-k-Lagrange elements to the computations of energy levels and wave functions of two-dimensional (2D) Bose-Einstein condensates (BEC), and BEC in a periodic potential. Sample numerical results are reported.