  We analyse the Virtual Element Methods (VEM) on a simple elliptic model
problem, allowing for more general meshes than the one typically considered in
the VEM literature. For instance, meshes with arbitrarily small edges (with
respect to the parent element diameter), can be dealt with. Our general
approach applies to different choices of the stability form, including, for
example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A.
Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual
element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214],
and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A
virtual element method for contact, submitted for publication]. Finally, we
show that the stabilization term can be simplified by dropping the contribution
of the internal-to-the-element degrees of freedom. The resulting stabilization
form, involving only the boundary degrees of freedom, can be used in the VEM
scheme without affecting the stability and convergence properties. The
numerical tests are in accordance with the theoretical predictions.

Numerical Analysis