On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

TL;DR

This paper reviews the construction and analysis of arbitrarily regular conforming virtual element methods for elliptic PDEs, emphasizing high regularity solutions and providing convergence results and space formulations.

Contribution

It introduces a general framework for high-regularity conforming VEM for elliptic problems, including space construction, degrees of freedom, and enhanced formulations.

Findings

Convergence in an energy norm is established.

Equivalent dimensions for regular and enhanced spaces are proven.

Practical examples for high regularity cases are provided.

Abstract

The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order $p_1$ using conforming, finite dimensional subspaces of $ H^{p_2}(\Omega)$, where $p_1$ and $p_2$ are two integer numbers such that $p_2 \geq p_1 \geq 1$ and $\Omega\in R^2$ is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects such as the choice and unisolvence of the degrees of freedom are discussed, also providing specific examples corresponding to various practical cases of high global regularity. Finally, the construction of the "enhanced" formulation of the virtual element spaces is also discussed in details with a proof that the dimension of the "regular" and "enhanced" spaces is the same and that the virtual element functions in both spaces can be described by the same choice of the degrees of freedom.