Lowest-order virtual element methods for linear elasticity problems

TL;DR

This paper introduces two lowest-order virtual element methods for planar linear elasticity, demonstrating their stability, convergence, and optimal rates, including in nearly incompressible cases.

Contribution

It develops two novel virtual element methods for linear elasticity, extending existing finite element ideas to the virtual element framework with proven stability and convergence.

Findings

Methods satisfy discrete Korn's inequality.

Converge uniformly for nearly incompressible cases.

Achieve optimal convergence rates.

Abstract

We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the virtual element method. We show that our proposed methods satisfy the discrete Korn's inequality. We also prove that the methods are convergent uniformly for the nearly incompressible case and the convergence rates are optimal.