scholarly journals The nonconforming Virtual Element Method for eigenvalue problems

2019 ◽  
Vol 53 (3) ◽  
pp. 749-774 ◽  
Author(s):  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca
Keyword(s):  
Eigenvalue Problems ◽  
Inner Product ◽  
Optimal Order ◽  
Large Set ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Discrete Eigenvalue ◽  
Order Error ◽  
Virtual Element ◽  
Element Method

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of theL2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

scholarly journals Virtual element method for quasilinear elliptic problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2450-2472 ◽  
Author(s):  
A Cangiani ◽  
P Chatzipantelidis ◽  
G Diwan ◽  
E H Georgoulis
Keyword(s):  
A Priori ◽  
Quasilinear Equation ◽  
Nonlinear Coefficient ◽  
Optimal Order ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Fixed Point Iterations ◽  
Quasilinear Elliptic ◽  
Element Method

Abstract A virtual element method for the quasilinear equation $-\textrm{div} ({\boldsymbol \kappa }(u)\operatorname{grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

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 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 virtual element method for the vibration problem of Kirchhoff plates

2018 ◽  
Vol 52 (4) ◽  
pp. 1437-1456 ◽  
Author(s):  
David Mora ◽  
Gonzalo Rivera ◽  
Iván Velásquez
Keyword(s):  
Thin Plates ◽  
Optimal Order ◽  
Transverse Displacement ◽  
Computational Domain ◽  
Virtual Element Method ◽  
Order Error ◽  
Vibration Problem ◽  
Kirchhoff Plates ◽  
Virtual Element ◽  
Element Method

The aim of this paper is to develop a virtual element method (VEM) for the vibration problem of thin plates on polygonal meshes. We consider a variational formulation relying only on the transverse displacement of the plate and propose anH2(Ω) conforming discretization by means of the VEM which is simple in terms of degrees of freedom and coding aspects. 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. Finally, we report several numerical experiments illustrating the behaviour of the proposed scheme and confirming our theoretical results on different families of meshes. Additional examples of cases not covered by our theory are also presented.

scholarly journals The virtual element method for eigenvalue problems with potential terms on polytopic meshes

2018 ◽  
Vol 63 (3) ◽  
pp. 333-365 ◽  
Author(s):  
Ondřej Čertík ◽  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca
Keyword(s):  
Eigenvalue Problems ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals A virtual element method for the von Kármán equations

2021 ◽  
Vol 55 (2) ◽  
pp. 533-560
Author(s):  
Carlo Lovadina ◽  
David Mora ◽  
Iván Velásquez
Keyword(s):  
Optimal Error Estimates ◽  
Transverse Displacement ◽  
Discrete Problem ◽  
Elastic Plates ◽  
Virtual Element Method ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Virtual Element ◽  
Von Kármán ◽  
Element Method

In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming inH2for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed forhsmall enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

scholarly journals A virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

2019 ◽  
Vol 40 (2) ◽  
pp. 1544-1576 ◽  
Author(s):  
Verónica Anaya ◽  
Mostafa Bendahmane ◽  
David Mora ◽  
Mauricio Sepúlveda
Keyword(s):  
Cardiac Electrophysiology ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Optimal Order ◽  
Computational Domain ◽  
Virtual Element Method ◽  
Compactness Criterion ◽  
Virtual Element ◽  
Element Method

AbstractWe present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an $H^1$-conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.

scholarly journals The nonconforming virtual element method for eigenvalue problems

2018 ◽  
Author(s):  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca
Keyword(s):  
Eigenvalue Problems ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals An H1-conforming virtual element for Darcy and Brinkman equations

2017 ◽  
Vol 28 (01) ◽  
pp. 159-194 ◽  
Author(s):  
Giuseppe Vacca
Keyword(s):  
Stokes Equations ◽  
Brinkman Equation ◽  
Optimal Order ◽  
Virtual Element Method ◽  
Element Space ◽  
Optimal Order Of Convergence ◽  
Brinkman Equations ◽  
Divergence Free ◽  
Virtual Element ◽  
Element Method

The focus of this paper is on developing a virtual element method (VEM) for Darcy and Brinkman equations. In [L. Beirão da Veiga, C. Lovadina and G. Vacca, ESAIM Math. Model. Numer. Anal. 51 (2017)], we presented a family of virtual elements for Stokes equations and we defined a new virtual element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different virtual element space having two fundamental properties: the [Formula: see text]-projection onto [Formula: see text] is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and [Formula: see text]-conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.

Sign in / Sign up

Export Citation Format

Share Document