The Shifted-Inverse Iteration Based on the Multigrid Discretizations for Eigenvalue Problems

Applied Numerical Mathematics

2016

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This paper proposes and analyzes an a posteriori error estimator for the finite element multi-scale discretization approximation of the Steklov eigenvalue problem. Based on the a posteriori error estimates, an adaptive algorithm of shifted inverse iteration type is designed. Finally, numerical experiments comparing the performances of three kinds of different adaptive algorithms are provided, which illustrate the efficiency of the adaptive algorithm proposed here.Keywords : Steklov eigenvalue problem, finite element, multi-scale discretization, adaptive algorithm, a posteriori error estimate.

Section: )mentioning

confidence: 96%

“…Our analysis is based on the following crucial property of the shifted-inverse iteration in finite element method (see Lemma 4.1 in [41]).…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem

Applied Numerical Mathematics

2016

Self Cite

“…Denote

dist (w, W) = \inf_{v \in W} | | w - v | |_{a}

. The following lemma is a crucial property of the shifted‐inverse iteration in finite element method (see Lemma 4.2 in ), which is a development of Theorem in . Lemma Suppose that (μ, u ) are the kth eigenpair of T and

μ_{h}

is the kth eigenvalue of T h .…”

Section: Multiscale Discretization Scheme Based On Rayleigh Quotient mentioning

confidence: 99%

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

Han

Zhang

Numerical Methods Partial

2014

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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

“…Xu and Zhou propose a two grid method based on inverse iteration for elliptic eigenvalue problems in [37]. Later, it's developed to multigrid method (e.g., see [22,27,34,38]), among which [27,34] establish a new type of multigrid scheme based on the multilevel correction. And it's successfully applied to selfadjoint Steklov eigenvalue problem [25,35], convecttion-diffusion eigenvalue problem [30], transmission eigenvalue problem [24], etc.…”

Section: Introductionmentioning

confidence: 99%

A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering

International Journal of Computer Mathematics

2019

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We propose a multigrid correction scheme to solve a new Steklov eigenvalue problem in inverse scattering. With this scheme, solving an eigenvalue problem in a fine finite element space is reduced to solve a series of boundary value problems in fine finite element spaces and a series of eigenvalue problems in the coarsest finite element space. And the coefficient matrices associated with those linear systems are constructed to be symmetric and positive definite. We prove error estimates of eigenvalues and eigenfunctions. Numerical results coincide in theoretical analysis and indicate our scheme is highly efficient in solving the eigenvalue problem.