Meiling Yue

Hehu Xie, Meiling Yue, Ning Zhang

This paper is concerned with the computable error estimates for the eigenvalue problem which is solved by the general conforming finite element methods on the general meshes. Based on the computable error estimate, we can give an asymptotically lower bound of the general eigenvalues. Furthermore, we also give a guaranteed upper bound of the error estimates for the first eigenfunction approximation and a guaranteed lower bound of the first eigenvalue based on computable error estimator. Some numerical examples are presented to validate the theoretical results deduced in this paper.

Hehu Xie, Fei Xu, Meiling Yue

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose-Einstein condensates based on Newton iteration method. Instead of treating eigenvalue $λ$ and eigenvector $u$ respectively, we regard the eigenpair $(λ, u)$ as one element in the composite space $\R \times H_0^1(Ω)$ and then Newton iteration method is adopted for the nonlinear problem. Thus in this multigrid scheme, we only need to solve a linear discrete boundary value problem in every refined space, which can improve the overall efficiency for the simulation of Bose-Einstein condensations.

Yunhui He, Qichen Hong, Hehu Xie et al.

This paper is to give a new understanding and applications of the subspace projection method for selfadjoint eigenvalue problems. A new error estimate in the energy norm, which is induced by the stiff matrix, of the subspace projection method for eigenvalue problems is given. The relation between error estimates in $L^2$-norm and energy norm is also deduced. Based on this relation, a new type of inverse power method is designed for eigenvalue problems and the corresponding convergence analysis is also provided. Then we present the analysis of the geometric and algebraic multigrid methods for eigenvalue problems based on the convergence result of the new inverse power method.