An Introduction to the Virtual Element Method

The Virtual Element Method (VEM) is a very recent technology for the discretization of partia differential equationsThe VEM can be interpreted as a novel approach that shares the same variational background as the Finite Element Method but enjoys also a connection with modern Mimetic schemes.

By avoiding the explicit integration of the shape functions that span the discrete Galerkin space and introducing a novel construction of the associated 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 solutions of arbitrary $C^k$ regularity, defined on unstructured meshes.

The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. After introducing the method on a simple model elliptic problem, we will present an (optimal) convergence result and some numerical tests. We will moreover address the practical construction of the scheme and also some interesting possibilities such as that of using high regularity discrete spaces and, if time allows, extension to (almost-incompressible) elasticity problems.

contact: marco.verani@polimi.it