In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

中文翻译：

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无发散有限元方法的连续内部惩罚稳定性

在本文中，我们针对不可压缩流体力学中的线性化问题，即具有低粘度的稳态 Oseen 方程，提出、分析和数值测试了压力鲁棒稳定有限元。稳定项由对流项的导数的不同组合在域三角剖分的元素面上的跳跃来定义。在这些稳定项的帮助下，以及假设有限元空间提供逐点无发散速度这一事实，$\mathcal O\big(h^{k+\frac 12}\big)$ 误差估计在 $L^2$ 范数中证明了该方法（在对流主导的体系中），并在误差的剩余范数中进行了最优阶估计。提供了支持理论发现的数值结果。