scholarly journals A Virtual Element Method for the Steklov Eigenvalue Problem Allowing Small Edges

2021 ◽  
Vol 88 (2) ◽  
Author(s):  
Felipe Lepe ◽  
David Mora ◽  
Gonzalo Rivera ◽  
Iván Velásquez
Keyword(s):  
Eigenvalue Problem ◽  
Virtual Element Method ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Virtual Element ◽  
Element Method

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.

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.

The Spectral Element Method for the Steklov Eigenvalue Problem

2013 ◽  
Vol 853 ◽  
pp. 631-635 ◽  
Author(s):  
Yan Jun Li ◽  
Yi Du Yang ◽  
Hai Bi
Keyword(s):  
Eigenvalue Problem ◽  
Spectral Element Method ◽  
A Priori ◽  
Spectral Element ◽  
Element Approximation ◽  
Steklov Eigenvalue ◽  
Steklov Eigenvalue Problem ◽  
Priori Error Estimates ◽  
Shaped Domain ◽  
Element Method

This paper discusses the spectral element approximation with LGL node basis for the Steklov eigenvalue problem, and analyzes the a priori error estimates. Finally, numerical experi-ments on the square and the L-shaped domain are carried out to get very accurate approximations by the spectral element method.

scholarly journals A virtual element method for the transmission eigenvalue problem

2018 ◽  
Vol 28 (14) ◽  
pp. 2803-2831 ◽  
Author(s):  
David Mora ◽  
Iván Velásquez
Keyword(s):  
Eigenvalue Problem ◽  
Optimal Order ◽  
Virtual Element Method ◽  
Order Error ◽  
Transmission Eigenvalue ◽  
Fourth Order Eigenvalue Problem ◽  
Transmission Eigenvalue Problem ◽  
Virtual Element ◽  
Adjoint Operators ◽  
Element Method

In this paper, we analyze a Virtual Element Method (VEM) for solving a non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. We write a variational formulation and propose a [Formula: see text]-conforming discretization by means of the VEM. We use the classical approximation theory for compact non-self-adjoint operators to obtain optimal order error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we present some numerical experiments illustrating the behavior of the virtual scheme on different families of meshes.

scholarly journals Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

Sign in / Sign up

Export Citation Format

Share Document