Abstract
AbstractA multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the multilevel correction for eigenvalue problems and the multigrid method for linear boundary value problems. In this scheme, obtaining the optimal approximation for the ground state solution of Bose-Einstein condensates includes a sequence of solutions of the linear boundary value problems by the multigrid method on the multilevel meshes and some solutions of nonlinear eigenvalue problems some very low dimensional finite element space. The total computational work of this scheme can reach almost the same optimal order as solving the corresponding linear boundary value problem. Therefore, this type of multigrid scheme can improve the overall efficiency for the simulation of Bose-Einstein condensations. Some numerical experiments are provided to validate the efficiency of the proposed method.
Highlights
Bose-Einstein condensation (BEC), which is a gas of bosons that are in the same quantum state, is an active field [6, 21, 27]
The aim of this paper is to present a multigrid scheme for Gross-Pitaevskii equation (GPE)
We propose a multigrid method to solve the GPE based on the multilevel correction method
Summary
Bose-Einstein condensation (BEC), which is a gas of bosons that are in the same quantum state, is an active field [6, 21, 27]. In 2001, the Nobel Prize in Physics was awarded Eric A. The properties of the condensate at zero or very low temperature [18, 29] can be described by the well-known Gross-Pitaevskii equation (GPE) [22, 26] which is a time-independent nonlinear Schrodinger equation [28]. It is found that the GPE fits well for the most of experiments [5, 16, 18, 24]
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