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Acta Numerica, 2023 The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very ...
Beirao Da Veiga, L +3 more
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Virtual element method for quasilinear elliptic problems [PDF]
IMA Journal of Numerical Analysis, 2019 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.
Cangiani A. +3 more
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The nonconforming virtual element method [PDF]
ESAIM: Mathematical Modelling and Numerical Analysis, 2016 We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods.
Ayuso De Dios, B +2 more
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The Virtual Element Method and its Applications
, 2022 The purpose of this book is to present the current state of the art of the Virtual Element Method (VEM) by collecting contributions from many of the most active researchers in this field and covering a broad range of topics: from the mathematical foundation to real life computational applications. The book is naturally divided into three parts.
Antonietti, PF +2 more
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The conforming virtual element method for polyharmonic problems [PDF]
Computers & Mathematics with Applications, 2020 In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form $Δ^p u =f$, $p\ge1$. More specifically, we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic ...
Antonietti P. F., Manzini G., Verani M.
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Interior estimates for the virtual element method
Numerische MathematikAbstractWe analyze the local accuracy of the virtual element method. More precisely, we prove an error bound similar to the one holding for the finite element method, namely, that the local $$H^1$$ H 1 error in a interior subdomain is bounded by a term ...
Silvia Bertoluzza +2 more
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BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS
Mathematical Models and Methods in Applied Sciences, 2012 We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach.
L Beirao da Veiga +5 more
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Stabilization of the nonconforming virtual element method [PDF]
Computers & Mathematics with Applications, 2022 We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the ...
S Bertoluzza +3 more
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The virtual element method for resistive magnetohydrodynamics [PDF]
Computer Methods in Applied Mechanics and Engineering, 2021 We present a virtual element method (VEM) for the numerical approximation of the electromagnetics subsystem of the resistive magnetohydrodynamics (MHD) model in two spatial dimensions. The major advantages of the virtual element method include great flexibility of polygonal meshes and automatic divergence-free constraint on the magnetic flux field.
S. Naranjo Alvarez +3 more
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The Neural Approximated Virtual Element Method for Elasticity Problems [PDF]
We present the Neural Approximated Virtual Element Method to numerically solve elasticity problems. This hybrid technique combines classical concepts from the Finite Element Method and the Virtual Element Method with recent advances in deep neural ...
Berrone, Stefano +2 more
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