PRIN 2017 - Virtual Element Methods: Analysis and Applications M.I.U.R. - MINISTERO DELL'ISTRUZIONE, DELL'UNIVERSITA' E DELLA RICERCA

P.F. Antonietti

M. Verani

August 2018

February 2023

The Virtual Element Method (VEM) is a novel technology for the discretization of Partial Differential Equations, that shares the same variational background as the Finite Element Method. By avoiding the explicit integration of the shape functions that span the discrete space and introducing an innovative construction of the stiffness matrices, the VEM acquires very interesting properties. The VEM easily allows for polygonal/polyhedral meshes also with non-convex elements; it allows for discrete spaces of arbitrary order k continuity on unstructured meshes; it allows to exactly enforce constraints on the discrete solution. The main scope of the project is to address the recent theoretical challenges posed by the extension of VEM to new and more complex problems and to assess whether this promising technology can achieve a breakthrough in various applications. On one side, the theoretical and computational foundations of VEM will be made stronger by investigating, for instance, robustness to geometry parameters and efficiency in terms of degrees of freedom. On the other side, we will focus on different problems of practical interest such as the development of VEM for Maxwell equations, polyharmonic problems, complex flows, elasto-plastic deformation problems and others.