Dr. Lourenco Beirao da Veiga
Professor of Numerical Analysis, Dipartimento di Matematica e Applicazioni
Università di Milano-Bicocca
The Virtual Element Method (VEM), is a very recent technology introduced in [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the discretization of partial differential equations, that has shared a good success in recent years. The VEM can be interpreted as a generalization of the Finite Element Method that allows to use general polygonal and polyhedral meshes, still keeping the same coding complexity and allowing for arbitrary degree of accuracy. The Virtual Element Method makes use of local functions that are not necessarily polynomials and are defined in an implicit way. Nevertheless, by a wise choice of the degrees of freedom and introducing a novel construction of the associated stiffness matrixes, the VEM avoids the explicit integration of such shape functions. In addition to the possibility to handle general polytopal meshes, the flexibility of the above construction yields other interesting properties with respect to more standard Galerkin methods. For instance, the VEM easily allows to build discrete spaces of arbitrary Ck regularity, or to satisfy exactly the divergence-free constraint for incompressible fluids. The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. We consider for simplicity a simple elliptic model problem (that is pure diffusion with variable coefficients) but set ourselves in the more involved 3D setting. In the first part we introduce the adopted Virtual Element space and the associated degrees of freedom, first by addressing the faces of the polyhedrons (i.e. polygons) and then building the space in the full volumes. We then describe the construction of the discrete bilinear form and the ensuing discretization of the problem. Furthermore, we show a set of theoretical and numerical results. In the very final part, we will give a glance at more involved problems, such as magnetostatics (mixed problem with more complex spaces interacting) and large deformation elasticity (nonlinear problem).
Dr. L. Beirao da Veiga obtained a PhD in Mathematics in 2005 at the University of Pavia, and is currently a full professor in the Math Department of the University of Milano-Bicocca. His scientific interests are in the range of numerical methods for PDEs, mainly the finite element method, isogeometric analysis, mimetic discretizations and virtual elements. He has more than 100 publications on international journals and a book on Mimetic Finite Difference (MFD) methods. In 2016 he was awarded the European Community on Computational Methods in Applied Sciences (ECCOMAS) Jacques Louis Lions Award and he is currently the PI of an ERC Consolidator Grant (2016-2021) focusing on Virtual Elements.