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Session Overview
plenary 4: Marco Verani
Tuesday, 13/July/2021:
3:00pm - 3:50pm

Session Chair: Claudio Canuto
Virtual location: Zoom 1

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Arbitrarily regular virtual element methods for elliptic partial differential equations

Marco Verani

Politecnico di Milano, Italy

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.

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