Numerical solution of partial differential equations
Numerical solution of partial differential equations
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation
Journal of Computational Physics
SIAM Journal on Scientific Computing
Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation
Journal of Computational Physics
Gauss-Seidel-type methods for energy states of a multi-component Bose-Einstein condensate
Journal of Computational Physics
Bose-Einstein Condensates and the Numerical Solution of the Gross-Pitaevskii Equation
Computing in Science and Engineering
Journal of Computational Physics
Journal of Computational Physics
A minimisation approach for computing the ground state of Gross-Pitaevskii systems
Journal of Computational Physics
High-order time-splitting Hermite and Fourier spectral methods
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Simulation of coherent structures in nonlinear Schrödinger-type equations
Journal of Computational Physics
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In this paper, we propose a new numerical method to compute the ground-state solution of trapped interacting Bose-Einstein condensation at zero or very low temperature by directly minimizing the energy functional via finite element approximation. As preparatory steps we begin with the 3d Gross-Pitaevskii equation (GPE), scale it to get a three-parameter model and show how to reduce it to 2d and 1d GPEs. The ground-state solution is formulated by minimizing the energy functional under a constraint, which is discretized by the finite element method. The finite element approximation for 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry are presented in detail and approximate ground-state solutions, which are used as initial guess in our practical numerical computation of the minimization problem, of the GPE in two extreme regimes: very weak interactions and strong repulsive interactions are provided. Numerical results in 1d, 2d with radial symmetry and 3d with spherical symmetry and cylindrical symmetry for atoms ranging up to millions in the condensation are reported to demonstrate the novel numerical method. Furthermore, comparisons between the ground-state solutions and their Thomas-Fermi approximations are also reported. Extension of the numerical method to compute the excited states of GPE is also presented.