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

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 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.

Virtual Element Method for the Navier--Stokes Equation coupled with the Heat Equation

“…Note that the heat equation for the temperature in (96) is convection-dominated. However, it is worth stressing that, differently form [38], we do not employ any stabilization technique to stabilize the discrete problem (we refer to [22,11,20] for the analysis of the stabilized method in the VEM context both for the convectiondominated elliptic equations and Oseen equations). Fig.…”

“…We refer to the book [10] where the current state of the art on the Virtual Element Method is reported, collecting contributions from many of the most active researchers in this field. VEMs for Stokes and Navier-Stokes equations have been proposed and analyzed in [9,15,44,16,13]; we refer, e.g., to [22,11,20], for the development of VEM for convection dominated problems.…”

Adaptive Virtual Element Method for Convection Dominated Diffusion Equations

“…Here the form b(• , •) is rewritten as the skew-symmetric part, which is a useful step in order to preserve the coercivity of A supg at the virtual discrete level, independently of the mesh size (see [7,18]).…”

“…Assumption 3.1 is only needed in the convection dominated case. In [7], under the above data assumption a robustness analysis of the SUPG-stabilized virtual elements for convection-diffusion problem is discussed and the optimal order of convergence is obtained, i.e.…”

“…This shows the efficiency of the SUPG stabilisation. Recently, Beirǎo da Veiga et al [7] discussed a robustness analysis of the SUPG-stabilized virtual elements for convection diffusion problems. By slightly modifying the SUPG format of [13], they propose a new way to discretize the convection term, which ultimately demonstrates the robustness of the parameters involved in the convergence estimates without requiring sufficiently small mesh sizes.…”