A review on arbitrarily regular conforming virtual element methods for elliptic partial differential equations

TL;DR

This paper reviews the development of arbitrarily regular conforming virtual element methods for elliptic PDEs, highlighting their mathematical foundation, construction of high-regularity spaces, and numerical performance analysis.

Contribution

It provides an abstract framework for high-regularity virtual element spaces and demonstrates their application to second- and fourth-order elliptic equations with numerical insights.

Findings

High-order continuity improves approximation accuracy.

Different stabilization strategies affect convergence rates.

Numerical experiments validate theoretical properties.

Abstract

The Virtual Element Method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order $2p_1$, for any integer $p_1\geq 1$. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of $H^{p_2}(\Omega)$, $\Omega$ being the computational domain and $p_2\geq p_1$ another suitable integer number. In this study, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order continuity on $\Omega$. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases $p_1=1$ and $2$, respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.