The talk will overview the different types of Virtual Element Methods that have been recently introduced in order to construct conforming approximations (continuous, H(curl)-conforming, H(div)-conforming, etc.) of the spaces commonly used for the numerical approximation of Boundary Value Problems for Partial Differential Equations. The most relevant feature of Virtual Element Methods is that the elements of VEM spaces are not made only of local polynomials, but include also solutions of local (systems of) PDE’s, that however do not need to be computed, not even in an approximate way. Some interesting effects of this approach is the possibility of using very general decompositions of the computational domain (say, into polygons or polyhedra) and also the possibility of using trial functions that satisfy exactly some interesting physical properties, as incompressibility, mass conservation, and so on.

Franco Brezzi works on the theoretical bases of Scientific Computing, and in particular in Numerical Methods for PDEs, where he made fundamental contributions to Mixed Finite Element Methods, Stabilization techniques, Mimetic Finite Differences, and Virtual Elements, with applications to Structural Mechanics, Fluid Mechanics, and Electro-Magnetics. He was awarded the Gauss-Newton gold medal of IACM at the World Congress of Computational Mechanics in 2004, the SIAM von Neumann award in 2009, and the Leonhard Euler Medal and the Ritz-Galerkin Medal of ECCOMAS in 2014 and 2016, respectively. Franco Brezzi was an Invited Plenary Speaker at the ICM in 2014 in Seoul, and an Invited Speaker at the ICM in 1986 in Berkeley. He had been President of the Italian Mathematical Union from 2006 to 2012, and Vice President of the European Mathematical Society from 2012 to 2016. He is also a member of the European Academy of Sciences, of the Academia Europaea, and of the Accademia dei Lincei, and fellow of IACM and of SIAM.