scholarly journals Basic convergence theory for the network element method

2021 ◽  
Author(s):  
Julien Coatléven
Keyword(s):  
Error Estimates ◽  
Numerical Scheme ◽  
Convergence Theory ◽  
Network Element ◽  
Basic Error ◽  
Virtual Element ◽  
Element Method

A recent paper introduced the network element method (NEM) where the usual mesh was replaced by a discretization network. Using the associated network geometric coefficients and following the virtual element framework, a consistent and stable numerical scheme was proposed. The aim of the present paper is to derive a convergence theory for the NEM under mild assumptions on the exact problem. We also derive basic error estimates, which are sub-optimal in the sense that we have to assume more regularity than usual.

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 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 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 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.

Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions

2018 ◽  
Vol 77 (3) ◽  
pp. 1874-1908 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals The virtual element method for resistive magnetohydrodynamics

2021 ◽  
Vol 381 ◽  
pp. 113815
Author(s):  
S. Naranjo Alvarez ◽  
V. Bokil ◽  
V. Gyrya ◽  
G. Manzini
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method
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