AN INTRODUCTION TO VIRTUAL ELEMENTS IN 3D

Series: 
Applied and Computational Mathematics Seminar
Monday, September 17, 2018 - 1:55pm
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Università di Milano-Bicocca
Organizer: 

This is a joint seminar by College of Engineering and School of Math.

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 C^k 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).