A Multigrid Method for Nonlinear Eigenvalue Problems

A multigrid method is proposed for solving nonlinear eigenvalue problems by the finite element method. With this new scheme, solving nonlinear eigenvalue problem is decomposed to a series of solutions of linear boundary value problems on multilevel finite element spaces and a series of small scale nonlinear eigenvalue problems. The computational work of this new scheme can reach almost the same as the solution of the corresponding linear boundary value problem. Therefore, this type of multilevel correction scheme improves the overfull efficiency of the nonlinear eigenvalue problem solving.Comment: 14 pages, 0 figure. arXiv admin note: text overlap with arXiv:1401.537

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

A 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 a series of solutions of nonlinear eigenvalue problems on the coarsest finite element space. The total computational work of this scheme can reach almost the optimal order as same 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.Comment: 15 pages and 6 figures. arXiv admin note: substantial text overlap with arXiv:1405.715

Asymptotic Expansions and Extrapolation of Approximate Eigenvalues for Second Order Elliptic Problems by Mixed Finite Element Methods

In this paper, we derive an asymptotic error expansion for the eigenvalue approximations by the lowest order Raviart-Thomas mixed finite element method for the general second order elliptic eigenvalue problems. Extrapolation based on such an expansion is applied to improve the accuracy of the eigenvalue approximations. Furthermore, we also prove the superclose property between the finite element projection with the finite element approximation of the eigenvalue problems by mixed finite element methods. In order to prove the full order of the eigenvalue extrapolation, we first propose "the auxiliary equation method". The result of this paper provides a general procedure to produce an asymptotic expansions for eigenvalue approximations by mixed finite elements.Comment: 13 pages, no figur

Postprocessing and Higher Order Convergence of Stabilized Finite Element Discretizations of the Stokes Eigenvalue Problem

In this paper, the stabilized finite element method based on local projection is applied to discretize the Stokes eigenvalue problems and the corresponding convergence analysis is given. Furthermore, we also use a method to improve the convergence rate for the eigenpair approximations of the Stokes eigenvalue problem. It is based on a postprocessing strategy that contains solving an additional Stokes source problem on an augmented finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of mixed finite element space. Numerical examples are given to confirm the theoretical analysis.Comment: 25 pages, 2 figure

A Full Multigrid Method for Eigenvalue Problems

In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use the multilevel correction method to transform the solution of eigenvalue problem to a series of solutions of the corresponding boundary value problems and eigenvalue problems defined on the coarsest finite element space. The boundary value problems which are define on a sequence of multilevel finite element space can be solved by some multigrid iteration steps. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as linear problem solver. The computational work of this new scheme can reach optimal order the same as solving the corresponding source problem. Therefore, this type of iteration scheme improves the efficiency of eigenvalue problem solving.Comment: 14vpages and 6 figures. arXiv admin note: substantial text overlap with arXiv:1409.2923, arXiv:1401.537

Computable Error Estimates for Ground State Solution of Bose-Einstein Condensates

In this paper, we propose a computable error estimate of the Gross-Pitaevskii equation for ground state solution of Bose-Einstein condensates by general conforming finite element methods on general meshes. Based on the proposed error estimate, asymptotic lower bounds of the smallest eigenvalue and ground state energy can be obtained. Several numerical examples are presented to validate our theoretical results in this paper.Comment: 19 pages, 9 figures. arXiv admin note: text overlap with arXiv:1601.0156

A Cascadic Multigrid Method for Eigenvalue Problem

A cascadic multigrid method is proposed for eigenvalue problems based on the multilevel correction scheme. With this new scheme, an eigenvalue problem on the finest space can be solved by smoothing steps on a series of multilevel finite element spaces and eigenvalue problem solving on the coarsest finite element space. Choosing the appropriate sequence of finite element spaces and the number of smoothing steps, the optimal convergence rate with the optimal computational work can be arrived. Some numerical experiments are presented to validate our theoretical analysis.Comment: 17 pages, 8 figure

Local a Priori Estimate on the General Scale Subdomains

The local a priori estimate for the finite element approximation is essential for underlying the local and parallel technique. It is well known that the constant coefficients in the inequality is independent of the mesh size. But it is not so clear whether the constant depends on the scale of the local subdomains. The aim of this note is to derive a new local a priori estimate on the general scale domains. We also show that the dependence of the constant appearing in the local a priori estimate on the scale of the subdomains.Comment: 11 pages, 0 figure

A Multilevel Correction Scheme for Nonsymmetric Eigenvalue Problems by Finite Element Methods

A multilevel correction scheme is proposed to solve defective and nodefective of nonsymmetric partial differential operators by the finite element method. The method includes multi correction steps in a sequence of finite element spaces. In each correction step, we only need to solve two source problems on a finer finite element space and two eigenvalue problems on the coarsest finite element space. The accuracy of the eigenpair approximation is improved after each correction step. This correction scheme improves overall efficiency of the finite element method in solving nonsymmetric eigenvalue problems.Comment: 17 pages, 16 figure

A Multilevel Correction Method for Interior Transmission Eigenvalue Problem

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can transform the transmission eigenvalue solving in the finest finite element space to a sequence of linear problems and some transmission eigenvalue solving in a very low dimensional spaces. Since the main computational work is to solve the sequence of linear problems, the multilevel correction method improves the overfull efficiency of the transmission eigenvalue solving. Some numerical examples are provided to validate the theoretical results and the efficiency of the proposed numerical scheme.Comment: 26 pages, 10 figures. arXiv admin note: text overlap with arXiv:1505.0628