A Space-time Adaptive Method For The Ground State Solution Of Bose-Einstein Condensate

Partial differential equations are widely used to describe many problems in modern science and engineering computation. It is critical to obtain the numerical solutions when we cannot solve the differential equations analytically. With the development of mathematical theory and com-putational methods, we can solve partial differential equations numerically by finite difference method, finite volume method and finite element method. However, when the solution is singular,it may cost too many computing resources using uniform mesh, especially the computation of high-dimensional problems exceeding the ability of computers. Moving mesh method redistributes mesh nodes according to the characteristics of the numerical solution. This method can reduce compu-tational error effectively without costing too many computational resources. In real computation,it may need long time to get the numerical solution using uniform time step. Temporal adaptive method adjust time steps automatically and it can improve computational efficiency.In 1925, Einstein predicted that the particles in the gas at a very low temperature would be in the same quantum state. In 1995, Bose-Einstein condensate(BEC) was first found in the di-lute weakly interacting gases. This problem draws the attention of physicists and mathematicians.The nonlinear Schrodinger(NLS) equation has been used extensively to describe the single particle properties of BECs. Many scholars have studied the nonlinear Schrodinger equations theoretically and numerically and proposed many numerical methods. With a box potential,there exist boundary layers in the ground state of BEC in the strongly repulsive interacting regime. Therefore, it may cost too many resources to compute the ground state of BEC using uniform mesh. The BEC ground state solution is a function found by minimizing the energy under a constraint. The energy changes fast at the beginning of computation, while it changes slowly when it is close to convergence. Ac-cording to the characteristics of numerical solution in space and time,it can improve computational efficiency using moving mesh method in space and temporal adaptive method in time.In this paper, we introduce a spatial and temporal adaptive finite element method to compute the ground state of Bose-Einstein condensates. First, this paper introduces the fundamentals of nonlinear Schrodinger equation and adaptive finite element method theoretically. Second, we in-troduce the 1D moving mesh method based on the equidistribution principle, the 2D moving mesh method based on harmonic mapping and temporal adaptive method. Then, we analyze the charac-teristics of the ground states in different cases and proposes the adaptive method in space and time respectively. Based on the spatial and temporal adaptive finite element method, we report some numerical results of one- and two-dimensional ground state solutions of BEC. Finally,we compare the numerical results in uniform mesh and moving mesh as well as temporal adaptive method and propose the advantages of spatial and temporal adaptive method.