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 Multilevel Correction Method for Steklov Eigenvalue Problem by Nonconforming Finite Element Methods

2015 ◽  
Vol 8 (3) ◽  
pp. 383-405 ◽  
Author(s):  
Xiaole Han ◽  
Yu Li ◽  
Hehu Xie
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Finite Element Methods ◽  
Correction Method ◽  
Nonconforming Finite Element ◽  
Element Space ◽  
Correction Scheme ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Nonconforming Finite Element Methods

AbstractIn this paper, a multilevel correction scheme is proposed to solve the Steklov eigenvalue problem by nonconforming finite element methods. With this new scheme, the accuracy of eigenpair approximations can be improved after each correction step which only needs to solve a source problem on finer finite element space and an Steklov eigenvalue problem on the coarsest finite element space. This correction scheme can increase the overall efficiency of solving eigenvalue problems by the nonconforming finite element method. Furthermore, as same as the direct eigenvalue solving by nonconforming finite element methods, this multilevel correction method can also produce the lower-bound approximations of the eigenvalues.

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.

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.

Steklov eigenvalue problem with a-harmonic solutions and variable exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Belhadj Karim ◽  
Abdellah Zerouali ◽  
Omar Chakrone
Keyword(s):  
Eigenvalue Problem ◽  
Smooth Boundary ◽  
Normal Vector ◽  
Variable Exponents ◽  
Variational Technique ◽  
Unit Normal Vector ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Outward Unit ◽  
Outward Unit Normal Vector

AbstractUsing the Ljusternik–Schnirelmann principle and a new variational technique, we prove that the following Steklov eigenvalue problem has infinitely many positive eigenvalue sequences:\left\{\begin{aligned} &\displaystyle\operatorname{div}(a(x,\nabla u))=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ &\displaystyle a(x,\nabla u)\cdot\nu=\lambda m(x)|u|^{p(x)-2}u&&\displaystyle% \phantom{}\text{on }\partial\Omega,\end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}}{(N\geq 2)} is a bounded domain of smooth boundary {\partial\Omega} and ν is the outward unit normal vector on {\partial\Omega}. The functions {m\in L^{\infty}(\partial\Omega)}, {p\colon\overline{\Omega}\mapsto\mathbb{R}} and {a\colon\overline{\Omega}\times\mathbb{R}^{N}\mapsto\mathbb{R}^{N}} satisfy appropriate conditions.

scholarly journals The Steklov Problem on Differential Forms

2019 ◽  
Vol 71 (2) ◽  
pp. 417-435
Author(s):  
Mikhail A. Karpukhin
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Properties ◽  
Differential Forms ◽  
Type Inequality ◽  
Slight Modification ◽  
Dimensional Manifold ◽  
Hodge Laplacian ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Dirichlet To Neumann Map

AbstractIn this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

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