The research activity on the numerical treatment of Partial Differential Equations (PDEs) using polygonal and polyhedral meshes has been mainly focused in the recent years on the development and analysis of the Mimetic Finite Difference (MFD) method, the Virtual Element Method (VEM), and the Discontinuous Galerkin (DG) Method. Each one of these three discretization methods provides a family of numerical schemes that are strictly related to the Finite Element Method (FEM), of which they can be considered different extensions and generalizations. All the schemes in these families are consistent in the sense that they are exact whenever the solution is a polynomial up to a given degree. This property determines the accuracy of the methods. Moreover, the stability of MFD and VEM is ensured by adding suitable stabilization terms that, for VEM, takes into consideration the non-polynomial part of the numerical solution. The domain of interest are diffusion problems (Poisson equation, Stokes equation), fluid dynamics and electromagnetics and the research activity is carried out through ongoing collaborations with the University of Pavia, IUSS-PV, University of Milano-Bicocca, University of Leicester (UK) and the Los Alamos National Laboratory (USA).
L. Beirão da Veiga, K. Lipnikov, and G. Manzini.
Arbitrary order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM Journal on Numerical Analysis 49(5):1737-1760, 2011.
L. Beirão da Veiga, F. Brezzi, C. Cangiani, G. Manzini, L. D. Marini, A. Russo.
Basic principles of the Virtual Element Method. Mathematical Models and Methods in Applied Sciences 23(1): 119--214, 2013
K. Lipnikov, G. Manzini, M. Shashkov.
The Mimetic Finite Difference Method. Journal of Computational Physics 257-Part B:1163-1227, 2014
K. Lipnikov, G. Manzini.
High-order mimetic methods for unstructured polyhedral meshes. Journal of Computational Physics 272:360-385, 2014