scholarly journals Convergence analysis of two finite element methods for the modified Maxwell's Steklov eigenvalue problem

2022 ◽  
Author(s):  
Bo Gong
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Convergence Analysis ◽  
Finite Element Methods ◽  
Electromagnetic Scattering ◽  
Scattering Problems ◽  
Steklov Eigenvalue ◽  
Monotonic Convergence ◽  
Steklov Eigenvalue Problem

The modified Maxwell's Steklov eigenvalue problem is a new problem arising from the study of inverse electromagnetic scattering problems. In this paper, we investigate two finite element methods for this problem and perform the convergence analysis. Moreover,  the monotonic convergence of the discrete eigenvalues computed by one of the methods is analyzed.

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 Conforming Finite Element Approximations for a Fourth-Order Steklov Eigenvalue Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Hai Bi ◽  
Shixian Ren ◽  
Yidu Yang
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Continuous Operator ◽  
Element Approximation ◽  
Morley Element ◽  
Steklov Eigenvalue ◽  
Conforming Finite Element ◽  
Steklov Eigenvalue Problem

This paper characterizes the spectrum of a fourth-order Steklov eigenvalue problem by using the spectral theory of completely continuous operator. The conforming finite element approximation for this problem is analyzed, and the error estimate is given. Finally, the bounds for Steklov eigenvalues on the square domain are provided by Bogner-Fox-Schmit element and Morley element.

A virtual element method for a biharmonic Steklov eigenvalue problem

2019 ◽  
Vol 10 (4) ◽  
pp. 325-337 ◽  
Author(s):  
Gabriel Monzón
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Fourth Order ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Virtual Element Method ◽  
Eigenvalues And Eigenfunctions ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Virtual Element

AbstractWe propose a finite element approximation for a fourth-order Steklov eigenvalue problem by means of the virtual elements. In this setting we derive error estimates for the eigenvalues and eigenfunctions under standard assumptions on the domain.

scholarly journals The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering

2020 ◽  
Vol 18 (1) ◽  
pp. 216-236
Author(s):  
Yu Zhang ◽  
Hai Bi ◽  
Yidu Yang
Keyword(s):  
Finite Element ◽  
Eigenvalue Problem ◽  
Inverse Scattering ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Upper And Lower Bounds ◽  
Adaptive Finite Element ◽  
Weak Formulation ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem

Abstract In this study, for the first time, we discuss the posteriori error estimates and adaptive algorithm for the non-self-adjoint Steklov eigenvalue problem in inverse scattering. The differential operator corresponding to this problem is non-self-adjoint and the associated weak formulation is not H 1-elliptic. Based on the study of Armentano et al. [Appl. Numer. Math. 58 (2008), 593–601], we first introduce error indicators for primal eigenfunction, dual eigenfunction, and eigenvalue. Second, we use Gårding’s inequality and duality technique to give the upper and lower bounds for energy norm of error of finite element eigenfunction, which shows that our indicators are reliable and efficient. Finally, we present numerical results to validate our theoretical analysis.

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