Virtual Element @ Bicocca
Recent developments in Virtual Elements at the
University of Milano - Bicocca
The Virtual Element Method (VEM, born in 2013) is a novel technology for the approximation of solutions to Partial Differential Equations (PDEs) that shares the same variational background as the Finite Element Method.
The VEM responds to the strongly increasing interest in using general polytopic meshes in the approximation of PDEs.
Avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the linear system stemming from the PDE discrerization, the VEM acquires very interesting properties and advantages over standard Galerkin methods, still retainging the same coding complexity.
For instance, the VEM easily allows for meshes (even nonconforming) possibly with non-convex elements and curved facets, and for discrete spaces of arbitrary C^k regularity.
The present page represents the research on the Virtual Element Method developed by the team at the Department of Mathematics and Applications (University of Milano-Bicocca), since the start up date of the ERC CoG project CAVE (year 2016).
A start-up package on Virtual Elements can be found in the following link.
It includes a pair of initial papers (the first seminal one and the one related to the coding of the scheme), and also the slides of the course on Virtual Elements held in June 2018 at the Dobbiaco Summer School. It also includes a MATLAB code for a simple model problem, which can be used as a starting point for newbies.
The main code of the team, Vem++, is written in C++ language and implements the VEM in two and three dimensions on general polytopic meshes. Several PDEs are therein discretized including diffusion, elasticity, Navier-Stokes, electro-magnetism problems.
Vem++ is freeware and available upon request; simply send an email to its developer Franco Dassi. Currently the code includes many tutorials that help to understand the overall structure of the code and how to plug-in new features. Furthermore, a short guide is available online. If you use Vem++, please cite the following reference in your work.
Other research interests
The group's interests are not limited to the virtual element method but also to numerical methods for PDEs in a broader sense and mesh generation/adaptation.
Several finite element, boundary element, and discontinuous Galerkin methods have been designed and analysed with focus on a wide spectrum of applications, including approximation of solutions to high convection fluido dynamics problem, higher-order problems, and magneto-hydrodynamics simulations, as well as the design of error estimators and parallel solvers.