The virtual element method

Abstract

The present review paper has several objectives. Its primary aim is to give an idea of the general features of virtual element methods (VEMs), which were introduced about a decade ago in the field of numerical methods for partial differential equations, in order to allow decompositions of the computational domain into polygons or polyhedra of a very general shape.Nonetheless, the paper is also addressed to readers who have already heard (and possibly read) about VEMs and are interested in gaining more precise information, in particular concerning their application in specific subfields such as ${C}^1$ -approximations of plate bending problems or approximations to problems in solid and fluid mechanics.

Divergence-Free Virtual Element Method for the Stokes Equations with Damping on Polygonal Meshes

“…Another crucial property of VEM is that the space of the basic functions is associated with the PDE problem inside the element, thus it is convenient to define divergence-free or rotation-free virtual elements [18]. For a thorough description of the VEM, see [3,19,41]. The (divergence-free) VEM are used to discretize the Stokes problem [12,24,29,40,42], the Brinkman problem [33,44,51], the Navier-Stokes problem [25,56] and so on [21,28,30,43,53].…”