Abstract: In recent years, there has been an intensive research on numerical approximations of partial differential equations on polygonal and polyhedral (polytopal, for short) meshes. Such research activity has led to the design of several families of numerical discretizations for PDEs, as, for example, the polygonal/polyhedral finite element method, the mimetic finite difference, the virtual element method, the discontinuous Galerkin method on polygonal/polyhedral grids, the hybrid discontinuous Galerkin method and the hybrid high-order method.
In this talk, we focus on the virtual element method (VEM) introduced in [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo 2013] which offers a great flexibility in designing approximation spaces featuring important properties other than just supporting polytopal meshes. The remarkable aspect that makes the VEM so appealing in this respect is that the formulation of arbitrarily regular approximations and their implementation are relatively straightforward. The crucial point here is that in the virtual element setting we do not need to know explicitly the shape functions spanning the virtual element space. The basis functions are uniquely defined by a set of values dubbed the degrees of freedom and these values are the only knowledge that are needed to formulate and implement the numerical scheme. During the talk, we show how this feature makes the construction of arbitrarily regular conforming virtual element approximations for linear elliptic equations of any order much simpler than, e.g., in the classical simplicial finite element context and almost immediate to implement. A priori error estimates in suitable norms and paradigmatic numerical examples will be also presented and discussed.