In this paper, we extend the divergence-free VEM of [L. Beirão da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier–Stokes problem on polygonal meshes, SIAM J. Numer. Anal. 56 (2018) 1210–1242] to the Oseen problem, including a suitable stabilization procedure that guarantees robustness in the convection-dominated case without disrupting the divergence-free property. The stabilization is inspired from [N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzman, A. Linke and C. Merdon, A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation, SIAM J. Numer. Anal. 59 (2021) 2746–2774] and includes local SUPG-like terms of the vorticity equation, internal jump terms for the velocity gradients, and an additional VEM stabilization. We derive theoretical convergence results that underline the robustness of the scheme in different regimes, including the convection-dominated case. Furthermore, as in the non-stabilized case, the influence of the pressure on the velocity error is moderate, as it appears only through higher-order terms.

中文翻译：

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Oseen 方程的涡量稳定虚拟元

在本文中，我们扩展了 [L. Beirão da Veiga、C. Lovadina 和 G. Vacca，多边形网格上 Navier-Stokes 问题的虚拟元素，暹罗学家编号。肛门。 56(2018) 1210–1242] 到 Oseen 问题，包括一个合适的稳定程序，在不破坏无散度特性的情况下保证对流主导情况下的鲁棒性。稳定性的灵感来自 [N. Ahmed、GR Barrenechea、E. Burman、J. Guzman、A. Linke 和 C. Merdon，在涡量方程中使用稳定性的 Oseen 方程的压力稳健离散化，暹罗学家编号。肛门。 59(2021) 2746–2774]，包括涡量方程的局部 SUPG 类项、速度梯度的内部跳跃项和额外的 VEM 稳定性。我们得出的理论收敛结果强调了该方案在不同状态下的稳健性，包括对流主导的情况。此外，在不稳定的情况下，压力对速度误差的影响是中等的，因为它只通过高阶项出现。