scholarly journals Some estimates of virtual element methods for fourth order problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qingguang Guan
Keyword(s):  
Error Estimates ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Regularity Estimates ◽  
Interpolation Operators ◽  
Virtual Element ◽  
The Right ◽  
Element Method

<p style='text-indent:20px;'>In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis bases on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates were obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions were proved.</p>

scholarly journals A posteriori error estimates for the virtual element method

2017 ◽  
Vol 137 (4) ◽  
pp. 857-893 ◽  
Author(s):  
Andrea Cangiani ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer ◽  
Oliver J. Sutton
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
Virtual Element Method ◽  
A Posteriori Error ◽  
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.

The Mixed Virtual Element Method for Grids with Curved Interfaces in Single-Phase Flow Problems

2021 ◽  
Author(s):  
Franco Dassi ◽  
Alessio Fumagalli ◽  
Davide Losapio ◽  
Stefano Scialò ◽  
Anna Scotti ◽  
...  
Keyword(s):  
Numerical Solutions ◽  
Industrial Applications ◽  
Velocity Fields ◽  
Three Dimensions ◽  
Darcy Law ◽  
Computational Domain ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

Abstract In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importance for the successful subsequent coupling with other physical phenomena such as transport. The virtual element method, as solution scheme, is known to be applicable in problems whose discretizations requires cells of general shape, and the mixed formulation is here preferred to obtain accurate velocity fields. To overcome the issues associated to the complex geometries and, at the same time, retaining the quality of the solutions, we present here the virtual element method to solve the Darcy problem, in mixed form, in presence of curved interfaces in two and three dimensions. The numerical scheme is presented in detail explaining the discrete setting with a focus on the treatment of curved interfaces. Examples, inspired from industrial applications, are presented showing the validity of the proposed approach.

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 The fully nonconforming virtual element method for biharmonic problems

2017 ◽  
Vol 28 (02) ◽  
pp. 387-407 ◽  
Author(s):  
P. F. Antonietti ◽  
G. Manzini ◽  
M. Verani
Keyword(s):  
Numerical Approximation ◽  
Optimal Error Estimates ◽  
Approximation Space ◽  
Virtual Element Method ◽  
Optimal Error ◽  
Order Of Accuracy ◽  
Element Discretization ◽  
Virtual Element ◽  
Biharmonic Problems ◽  
Element Method

In this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic problems. The approximation space is made of possibly discontinuous functions, thus giving rise to the fully nonconforming virtual element method. We derive optimal error estimates in a suitable (broken) energy norm and present numerical results to assess the validity of the theoretical estimates.

scholarly journals Equilibrium analysis of an immersed rigid leaflet by the virtual element method

2021 ◽  
pp. 1-50
Author(s):  
L. Beirão da Veiga ◽  
C. Canuto ◽  
R. H. Nochetto ◽  
G. Vacca
Keyword(s):  
Numerical Technique ◽  
Angular Position ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Bisection Algorithm ◽  
Computational Mesh ◽  
Virtual Element ◽  
Element Method

We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the properties of the domain functional describing the angular momentum exerted by the fluid on the leaflet (which depends on both the leaflet angular position and its thickness), we identify sufficient conditions on the spring stiffness function for the existence (and uniqueness) of equilibrium positions. This study resorts to techniques from shape differential calculus. We propose a numerical technique that exploits the mesh flexibility of the Virtual Element Method (VEM). A (polygonal) computational mesh is generated by cutting a fixed background grid with the leaflet geometry, and the problem is then solved with stable VEM Stokes elements of degrees 1 and 2 combined with a bisection algorithm. We prove quasi-optimal error estimates and present a large array of numerical experiments to document the accuracy and robustness with respect to degenerate geometry of the proposed methodology.

Sign in / Sign up

Export Citation Format

Share Document