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The objective of this contribution is to develop a convergence analysis forSUPG-stabilized Virtual Element Methods in diffusion-convection problems thatis robust also in the convection dominated regime. For the original methodintroduced in [Benedetto et al, CMAME 2016] we are able to show an "almostuniform" error bound (in the sense that the unique term that depends in anunfavorable way on the parameters is damped by a higher order mesh-sizemultiplicative factor). We also introduce a novel discretization of theconvection term that allows us to develop error estimates that are fully robustin the convection dominated cases. We finally present some numerical result.

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