SUPG-stabilized Virtual Elements for diffusion-convection problems: a robustness analysis

TL;DR

This paper develops a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems, achieving robustness in convection-dominated regimes through novel discretization and error bounds.

Contribution

It introduces a new discretization of the convection term and provides a robust convergence analysis for SUPG-stabilized Virtual Elements in convection-dominated problems.

Findings

Almost uniform error bounds achieved.

New discretization ensures robustness in convection-dominated regimes.

Numerical results confirm theoretical robustness.

Abstract

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 unfavorable 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.