Recent results and perspectives for virtual element methods

This editorial paper is devoted to present the papers published in a special issue focused on recent views, applications, and results of virtual element methods (VEM). A critical analysis of the overall contents of the issue is proposed, thus leading to a forward look to research perspectives.

SUPG-stabilized virtual elements for diffusion-convection problems: a robustness analysis

The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an “almost uniform” error bound (in the sense that the unique term that depends in an unfavourable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.