  New efficient and accurate numerical methods are proposed to compute ground
states and dynamics of dipolar Bose-Einstein condensates (BECs) described by a
three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a dipolar
interaction potential. Due to the high singularity in the dipolar interaction
potential, it brings significant difficulties in mathematical analysis and
numerical simulations of dipolar BECs. In this paper, by decoupling the
two-body dipolar interaction potential into short-range (or local) and
long-range interactions (or repulsive and attractive interactions), the GPE for
dipolar BECs is reformulated as a Gross-Pitaevskii-Poisson type system. Based
on this new mathematical formulation, we prove rigorously existence and
uniqueness as well as nonexistence of the ground states, and discuss the
existence of global weak solution and finite time blowup of the dynamics in
different parameter regimes of dipolar BECs. In addition, a backward Euler sine
pseudospectral method is presented for computing the ground states and a
time-splitting sine pseudospectral method is proposed for computing the
dynamics of dipolar BECs. Due to the adaption of new mathematical formulation,
our new numerical methods avoid evaluating integrals with high singularity and
thus they are more efficient and accurate than those numerical methods
currently used in the literatures for solving the problem.
  Extensive numerical examples in 3D are reported to demonstrate the efficiency
and accuracy of our new numerical methods for computing the ground states and
dynamics of dipolar BECs.
