The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

Li, Hao; Yang, Yidu

doi:10.1016/j.camwa.2013.01.043

Cited by 15 publications

(19 citation statements)

References 21 publications

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“…Superconvergence result O(N −1 ) and ultraconvergence O(N −2 ) are numerically observed for eigenfunction and eigenvalue approximation respectively. However, methods in [23,41] can only numerically give asymptotically optimal results. We want to emphasize that the new algorithms can get superconvergence or ultraconvergence results with no more or even less computational cost compared to the methods proposed in [23,41].…”

Section: )mentioning

confidence: 99%

“…Compared to methods in [23,41], Algorithm 3 and 4 use recovery based a posteriori error estimator. The propose of gradient recovery in the above two algorithms is twofold.…”

Section: )mentioning

confidence: 99%

“…Using our methods, solving an eigenvalue problem by AFEM will not be more difficult than solving a boundary value problem by AFEM. The most important feature which distinguishes them from the methods in [23,41] is that superconvergence of eigenfunction approximation and ultraconvergence (two order higher) of eigenvalue approximation can be numerically observed.…”

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confidence: 99%

“…To reduce computational cost, Mehrmann and Miedlar [30] introduced a new adaptive method which only requires an inexact solution of algebraic eigenvalue equation on each iteration by only performing a few iterations of Krylov subspace solver. Recently, Li and Yang [23] proposed an adaptive finite element method based on multi-scale discretization for eigenvalue problems and Xie [41] introduced a type of adaptive finite element method based on the multilevel correction scheme. Both methods only solve an eigenvalue problem on the coarsest mesh and solve boundary value problems on adaptive refined meshes.…”

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confidence: 99%

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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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Section: )mentioning

confidence: 99%

“…Compared to methods in [23,41], Algorithm 3 and 4 use recovery based a posteriori error estimator. The propose of gradient recovery in the above two algorithms is twofold.…”

Section: )mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

2016

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“…In particular, Hu and Cheng [14] proposed an accelerated two-grid discretization scheme for solving elliptic eigenvalue problems. Yang et al [28] presented a two-grid discretization scheme based on shifted-inverse power method for elliptic eigenvalue problems and then discussed the adaptive finite element method based on multi-scale discretization for the eigenvalue problems in [19]. The two-grid method for the second order elliptic problems by mixed finite element method has been established in [8], [24].…”

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confidence: 99%

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

2014

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A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem

Han

Zhang

Yang

2014

Numerical Methods Partial

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In this article, we combine mixed finite element method, multiscale discretization, and Rayleigh quotient iteration to propose a new adaptive algorithm based on residual type a posterior error estimates for the Stokes eigenvalue problem. Both reliability and efficiency of the error indicator are proved. The efficiency of the algorithm is also investigated using Chen's Innovation Finite Element Method (iFEM) package. Numerical results are satisfying.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 31–53, 2015

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The adaptive finite element method based on multi-scale discretizations for eigenvalue problems

Cited by 15 publications

References 21 publications

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

Superconvergent Two-Grid Methods for Elliptic Eigenvalue Problems

Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem

A new adaptive mixed finite element method based on residual type a posterior error estimates for the Stokes eigenvalue problem

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