Spectral collocation and a two-level continuation scheme for dipolar Bose-Einstein condensates

We exploit the high accuracy of spectral collocation methods in the context of a two-level continuation scheme for computing ground state solutions of dipolar Bose-Einstein condensates (BEC), where the first kind Chebyshev polynomials and Fourier sine functions are used as the basis functions for the trial function space. The governing Gross-Pitaevskii equation (or Schrödinger equation) can be reformulated as a Schrödinger-Poisson (SP) type system [13]. The two-level continuation scheme is developed for tracing the first solution curves of the SP system, which in turn provide an appropriate initial guess for the Newton method to compute ground state solutions for 3D dipolar BEC. Extensive numerical experiments on 3D dipolar BEC and dipolar BEC in optical lattices are reported.