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Challenges and Advancements in Virtual Elements

Total Cost €


EC-Contrib. €






 CAVE project word cloud

Explore the words cloud of the CAVE project. It provides you a very rough idea of what is the project "CAVE" about.

data    stronger    tough    space    regularity    efficient    convex    mri    posed    computational    exact    unstructured    spaces    construction    bidomain    keeping    instance    interesting    de    galerkin    deformation    discretization    handling    easily    standard    possibly    limit    potentials    plus    gain    satisfaction    yield    pdes    deeper    polyhedral    laws    robustness    vem    front    span    flexible    approximation    background    applicative    distortions    foundations    theoretical    explicit    practical    matrixes    accurate    grids    conforming    numerical    dramatically    domain    shares    stiffness    shape    innovative    partial    cardiac    date    procedure    meshing    model    elasticity    curved    hexahedral    introducing    made    variational    advantages    virtual    polygonal    breakthrough    faces    grid    conservation    responds    propagation    inclusions    avoiding    equations    meshes    discrete    first    complexity    integration    material    differential    functions    refinement    acquires    finite    scope    respect    adaptivity    benchmark    purposes    coding    arbitrary    tetrahedral      

Project "CAVE" data sheet

The following table provides information about the project.


Organization address
city: MILANO
postcode: 20126

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country Italy [IT]
 Total cost 980˙634 €
 EC max contribution 980˙634 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-CoG
 Funding Scheme ERC-COG
 Starting year 2016
 Duration (year-month-day) from 2016-07-01   to  2021-06-30


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

The Virtual Element Method (VEM) is a novel technology for the discretization of partial differential equations (PDEs), that shares the same variational background as the Finite Element Method. First but not only, the VEM responds to the strongly increasing interest in using general polyhedral and polygonal meshes in the approximation of PDEs without the limit of using tetrahedral or hexahedral grids. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrixes, the VEM acquires very interesting properties and advantages with respect to more standard Galerkin methods, yet still keeping the same coding complexity. For instance, the VEM easily allows for polygonal/polyhedral meshes (even non-conforming) with non-convex elements and possibly with curved faces; it allows for discrete spaces of arbitrary C^k regularity on unstructured meshes. The main scope of the project is to address the recent theoretical challenges posed by VEM and to assess whether this promising technology can achieve a breakthrough in applications. First, the theoretical and computational foundations of VEM will be made stronger. A deeper theoretical insight, supported by a wider numerical experience on benchmark problems, will be developed to gain a better understanding of the method's potentials and set the foundations for more applicative purposes. Second, we will focus our attention on two tough and up-to-date problems of practical interest: large deformation elasticity (where VEM can yield a dramatically more efficient handling of material inclusions, meshing of the domain and grid adaptivity, plus a much stronger robustness with respect to large grid distortions) and the cardiac bidomain model (where VEM can lead to a more accurate domain approximation through MRI data, a flexible refinement/de-refinement procedure along the propagation front, to an exact satisfaction of conservation laws).


year authors and title journal last update
List of publications.
2019 L. Beirao da Veiga, G. Manzini, L. Mascotto
A posteriori error estimation and adaptivity in hp virtual elements
published pages: , ISSN: 0945-3245, DOI: 10.1007/s00211-019-01054-6
Numerische Mathematik 2019-10-15
2017 L. Beirão da Veiga, A. Chernov, L. Mascotto, A. Russo
Exponential convergence of the hp virtual element method in presence of corner singularities
published pages: 581–613, ISSN: 0029-599X, DOI: 10.1007/s00211-017-0921-7
Numerische Mathematik 138 2019-10-15
2017 L. Beirão da Veiga, F. Dassi, A. Russo
High-order Virtual Element Method on polyhedral meshes
published pages: 1110-1122, ISSN: 0898-1221, DOI: 10.1016/j.camwa.2017.03.021
Computers & Mathematics with Applications 74/5 2019-10-15
2017 L. Beirão da Veiga, F. Brezzi, F. Dassi, L.D. Marini, A. Russo
Virtual Element approximation of 2D magnetostatic problems
published pages: 173-195, ISSN: 0045-7825, DOI: 10.1016/j.cma.2017.08.013
Computer Methods in Applied Mechanics and Engineering 327 2019-10-15
2017 Lourenço Beirão da Veiga, Carlo Lovadina, Alessandro Russo
Stability analysis for the virtual element method
published pages: 2557-2594, ISSN: 0218-2025, DOI: 10.1142/S021820251750052X
Mathematical Models and Methods in Applied Sciences 27/13 2019-10-15
2019 Lourenço Beirão da Veiga, Alessandro Russo, Giuseppe Vacca
The Virtual Element Method with curved edges
published pages: , ISSN: 0764-583X, DOI: 10.1051/m2an/2018052
ESAIM: Mathematical Modelling and Numerical Analysis 2019-10-15
2018 L. Beirão da Veiga, F. Brezzi, F. Dassi, L.D. Marini, A. Russo
Lowest order Virtual Element approximation of magnetostatic problems
published pages: 343-362, ISSN: 0045-7825, DOI: 10.1016/j.cma.2017.12.028
Computer Methods in Applied Mechanics and Engineering 332 2019-10-15
2018 L. Beira͂o da Veiga, C. Lovadina, G. Vacca
Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes
published pages: 1210-1242, ISSN: 0036-1429, DOI: 10.1137/17m1132811
SIAM Journal on Numerical Analysis 56/3 2019-10-15
2018 Lourenço Beirão da Veiga, Franco Brezzi, L. Donatella Marini, Alessandro Russo
Virtual Element approximations of the Vector Potential Formulation of Magnetostatic problems
published pages: 399-416, ISSN: 2426-8399, DOI: 10.5802/smai-jcm.40
SMAI Journal of Computational Mathematics 4 2019-10-15
2017 H. Chi, L. Beirão da Veiga, G.H. Paulino
Some basic formulations of the virtual element method (VEM) for finite deformations
published pages: 148-192, ISSN: 0045-7825, DOI: 10.1016/j.cma.2016.12.020
Computer Methods in Applied Mechanics and Engineering 318 2019-10-15

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