Gauss–Seidel-type methods for energy states of a multi-component Bose–Einstein condensate

In this paper, we propose two iterative methods, a Jacobi-type iteration (JI) and a Gauss–Seidel-type iteration (GSI), for the computation of energy states of the time-independent vector Gross–Pitaevskii equation (VGPE) which describes a multi-component Bose–Einstein condensate (BEC). A discretization of the VGPE leads to a nonlinear algebraic eigenvalue problem (NAEP). We prove that the GSI method converges locally and linearly to a solution of the NAEP if and only if the associated minimized energy functional problem has a strictly local minimum. The GSI method can thus be used to compute ground states and positive bound states, as well as the corresponding energies of a multi-component BEC. Numerical experience shows that the GSI converges much faster than JI and converges globally within 10–20 steps.