Basic principles of mixed Virtual Element Methods

Abstract

Abstract. The aim of this paper is to give a simple, introductory presentation of the extension of the Virtual Element Method to the discretization of H(div)-conforming vector fields (or, more generally, of (n − 1) − Cochains). As we shall see, the methods presented here can be seen as extensions of the so-called BDM family to deal with more general element geometries (such as polygons with an almost arbitrary geometry). For the sake of simplicity, we limit ourselves to the 2-dimensional case, with the aim of making the basic philosophy clear. However, we consider an arbitrary degree of accuracy k (the Virtual Element analogue of dealing with polynomials of arbitrary order in the Finite Element Framework).Mathematics Subject Classification. 65N30, 65N12, 65N15, 76R50.

Virtual elements for a shear-deflection formulation of Reissner–Mindlin plates

“…Now, we bound the second term on the right hand side of (32) and we follow similar steps as in Lemma 4.1 to derive (20). In fact, using the definition (17), and adding and subtracting g h we rewrite the term as follows…”

“…Indeed, by avoiding the explicit construction of the local basis functions, the VEM can easily handle general polygons/polyhedrons without complex integrations on the element (see [9] for details on the coding aspects of the method). The Virtual Element Method has been applied successfully in a large range of problems, see for instance [1,2,7,8,9,12,15,16,17,20,23,25,28,35,39,40,41,47,48].…”

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

“…First introduced in [4] and extended in [5,6,3,21,2,26], the Virtual Element Method allows the use of quite general non-degenerate and star-shaped polygons to mesh the spatial domain, even including the possibility of straight angles. In the present framework, we take advantage from this flexibility to easily build a mesh which, on each fracture, is locally or globally conforming to the traces.…”

The nonconforming virtual element method

“…Despite of its infancy, the conforming VEM laid in [9] has been already extended to a variety of two dimensional problems: plate problems are studied in [16], linear elasticity in [10], mixed methods for H(div; Ω)-approximations are introduced in [15], and very recently the VEM has been extended to simulations on discrete fracture networks [11]. In [2], further tools are presented that allow us to construct and analyze the conforming VEM for three dimensional elliptic problems.…”