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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.

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

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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