A combined discontinuous Galerkin method for the dipolar Bose-Einstein condensation

In this work, a combined discontinuous Galerkin (DG) method, which is a hybridized mixed discontinuous Galerkin (HMDG) method combined with the direct discontinuous Galerkin (DDG) method, is proposed to compute ground states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a multi-dimensional Gross-Pitaevskii equation (GPE) coupled with a first-order velocity system. Due to the adaption of the first-order velocity system instead of dipolar interactions, the proposed combined DG method avoids to evaluate integrals with high singularity. Additionally, this method keeps the conservation of the particle number. The Krylov semi-implicit method is applied to the time discretization. Finally, numerical examples are presented to demonstrate the accuracy and capability of the proposed method.