scholarly journals Multiscale Discretization Scheme Based on the Rayleigh Quotient Iterative Method for the Steklov Eigenvalue Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Hai Bi ◽  
Yidu Yang
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problem ◽  
Iterative Method ◽  
Error Estimates ◽  
Adaptive Algorithm ◽  
Numerical Experiments ◽  
Rayleigh Quotient ◽  
Discretization Scheme ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

This paper discusses efficient numerical methods for the Steklov eigenvalue problem and establishes a new multiscale discretization scheme and an adaptive algorithm based on the Rayleigh quotient iterative method. The efficiency of these schemes is analyzed theoretically, and the constants appeared in the error estimates are also analyzed elaborately. Finally, numerical experiments are provided to support the theory.

A Two-Grid Method of Nonconforming Element Based on the Shifted-Inverse Power Method for the Steklov Eigenvalue Problem

2013 ◽  
Vol 694-697 ◽  
pp. 2918-2921
Author(s):  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
High Efficiency ◽  
Grid Method ◽  
Inverse Power ◽  
Power Method ◽  
Discretization Scheme ◽  
Nonconforming Element ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Numer Anal

This paper establishes a new kind of two-grid discretization scheme of nonconforming Crouzeix-Raviart element based on the shifted-inverse power method for the Steklov eigenvalue problem. The error estimates are provided from the work of Yang and Bi (SIAM J. Numer. Anal., 49, pp.1602-1624, 2011). Finally, numerical experiments are reported to illustrate the high efficiency of the two-grid discretization scheme proposed in this paper.

scholarly journals A Multilevel Correction Scheme for the Steklov Eigenvalue Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Qichao Zhao ◽  
Yidu Yang ◽  
Hai Bi
Keyword(s):  
Theoretical Analysis ◽  
Eigenvalue Problem ◽  
Numerical Experiments ◽  
Inverse Iteration ◽  
Correction Technique ◽  
Correction Scheme ◽  
Multiple Eigenvalues ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

Combining the correction technique proposed by Lin and Xie and the shifted inverse iteration, a multilevel correction scheme for the Steklov eigenvalue problem is proposed in this paper. The theoretical analysis and numerical experiments indicate that the scheme proposed in this paper is efficient for both simple and multiple eigenvalues of the Steklov eigenvalue problem.

A Two-Grid Discretization Scheme for a Sort of Steklov Eigenvalue Problem

2012 ◽  
Vol 557-559 ◽  
pp. 2087-2091
Author(s):  
Chao Xia ◽  
Yi Du Yang ◽  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
Algebraic System ◽  
Discretization Scheme ◽  
Linear Algebraic System ◽  
Fine Grid ◽  
Asymptotically Optimal ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Optimal Accuracy ◽  
Numer Anal

On the basis of Yang and Bi’s work (SIAM J Numer Anal 49, p.1602-1624), this paper discusses a discretization scheme for a sort of Steklov eigenvalue problem and proves the high effiency of the scheme. With the scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid. And the resulting solution can maintain an asymptotically optimal accuracy. Finally, the numerical results are provided to support the theoretical analysis..

scholarly journals A virtual element method for the Steklov eigenvalue problem

2015 ◽  
Vol 25 (08) ◽  
pp. 1421-1445 ◽  
Author(s):  
David Mora ◽  
Gonzalo Rivera ◽  
Rodolfo Rodríguez
Keyword(s):  
Eigenvalue Problem ◽  
Error Estimates ◽  
Optimal Order ◽  
Computational Domain ◽  
Virtual Element Method ◽  
Order Error ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Virtual Element ◽  
Element Method

The aim of this paper is to develop a virtual element method for the two-dimensional Steklov eigenvalue problem. We propose a discretization by means of the virtual elements presented in [L. Beirão da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci.23 (2013) 199–214]. Under standard assumptions on the computational domain, we establish that the resulting scheme provides a correct approximation of the spectrum and prove optimal-order error estimates for the eigenfunctions and a double order for the eigenvalues. We also prove higher-order error estimates for the computation of the eigensolutions on the boundary, which in some Steklov problems (computing sloshing modes, for instance) provides the quantity of main interest (the free surface of the liquid). Finally, we report some numerical tests supporting the theoretical results.

scholarly journals A Type of Multigrid Method Based on the Fixed-Shift Inverse Iteration for the Steklov Eigenvalue Problem

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Feiyan Li ◽  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Inverse Iteration ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.

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