A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems

Chien, C.-S.; Jeng, B.-W.

doi:10.1137/030602447

Cited by 49 publications

(23 citation statements)

References 29 publications

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“…The idea of the two-grid method is related to the ideas in [23,24] for nonsymmetric or indefinite problems and nonlinear elliptic equations. Since then, many numerical 1 methods for solving eigenvalue problems based on the idea of the two-grid method are developed (see, e.g., [5,12,14,28,34,42]). A type of multi-level correction scheme is presented by LinXie [33] and Xie [40].…”

Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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Abstract. In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization.Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

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Section: Introductionmentioning

confidence: 99%

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Zhang

2017

J Sci Comput

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“…Comp, 70(2001) To reduce the computational cost of eigenvalue problems, Xu and Zhou introduced a two-grid discretization scheme [42]. Later on, similar ideas were applied to non self-adjoint eigenvalue problems [22] and semilinear elliptic eigenvalue problems [11]. Furthermore, it also has been generalized to three-scale discretization [16] and multilevel discretization [24].…”

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confidence: 99%

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

2016

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Abstract. Some numerical algorithms for elliptic eigenvalue problems are proposed, analyzed, and numerically tested. The methods combine advantages of the two-grid algorithm [J. Xu and A. Zhou, Math. Comp, 70(2001) To reduce the computational cost of eigenvalue problems, Xu and Zhou introduced a two-grid discretization scheme [42]. Later on, similar ideas were applied to non self-adjoint eigenvalue problems [22] and semilinear elliptic eigenvalue problems [11]. Furthermore, it also has been generalized to three-scale discretization [16] and multilevel discretization [24]. Recently, a new shifted-inverse power method based two-grid scheme was proposed in [21,43].To improve accuracy of eigenvalue approximation, many methods have been proposed. In [37], Shen and Zhou introduced a defect correction scheme based on averaging recovery, like a global L 2 projection and a Clément-type operator. In [34], Naga, Zhang, and Zhou used Polynomial Preserving Recovery to enhance eigenvalue approximation. In [40], Wu and Zhang further showed polynomial preserving recovery can even enhance eigenvalue approximation on adaptive meshes. The idea was further studied in [31,15]. Alternatively, Racheva and Andreev proposed a two-space method to achieve better eigenvalue approximation [36] and it was also applied to biharmonic eigenvalue problem [1].In this paper, we propose some fast and efficient solvers for elliptic eigenvalue problems. We combine ideas of the two-grid method, two-space method, shiftedinverse power method, and PPR recovery enhancement to design our new algorithms. The first purpose is to introduce two superconvergent two-grid methods for eigenvalue problems. Our first algorithm is a combination of the shifted-inverse power based two-

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“…Hence, it can be seen as a postprocessing technique and can take less CPU time compared to the one grid methods. To the best of our knowledge, some details of the two-grid scheme can be found in the works of Xu and Zhou [13], Chien and Jeng [14,15], Chen et al [7,16], Hu and Cheng [17], Yang et al [18,19], Huang et al [8], and Weng et al [20,21]. The two-space method is actually the iterative Galerkin method, which was first used for solving integral equation eigenvalue problems by Sloan [22] and differential equation eigenvalue problems by Lin and Xie [23].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

Weng

Cai

2017

Mathematical Problems in Engineering

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This paper provides a two-space stabilized mixed finite element scheme for the Stokes eigenvalue problem based on local Gauss integration. The two-space strategy contains solving one Stokes eigenvalue problem using the 1 − 1 finite element pair and then solving an additional Stokes problem using the 2 − 2 finite element pair. The postprocessing technique which increases the order of mixed finite element space by using the same mesh can accelerate the convergence rate of the eigenpair approximations. Moreover, our method can save a large amount of computational time and the corresponding convergence analysis is given. Finally, numerical results are presented to confirm the theoretical analysis.

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A Two-Grid Discretization Scheme for Semilinear Elliptic Eigenvalue Problems

Cited by 49 publications

References 29 publications

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

A Multi-Level Mixed Element Method for the Eigenvalue Problem of Biharmonic Equation

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

An Efficient Algorithm with Stabilized Finite Element Method for the Stokes Eigenvalue Problem

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