scholarly journals A Multigrid Method for Ground State Solution of Bose-Einstein Condensates

2016 ◽  
Vol 19 (3) ◽  
pp. 648-662 ◽  
Author(s):  
Hehu Xie ◽  
Manting Xie
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Ground State ◽  
Multigrid Method ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Ground State Solution ◽  
State Solution ◽  
Linear Boundary ◽  
Bose Einstein

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.

scholarly journals A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

2021 ◽  
Author(s):  
Manting Xie ◽  
Fei Xu ◽  
Meiling Yue
Keyword(s):  
Finite Element ◽  
Boundary Value Problems ◽  
Multigrid Method ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Positive Definite ◽  
Adaptive Finite Element Method ◽  
Steklov Eigenvalue ◽  
Full Multigrid Method ◽  
Steklov Eigenvalue Problems

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

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