SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page A2614-A2637, January 2021.
In this paper, we propose a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence on polytopal mesh in the primary velocity-pressure formulation. Convergence rates with one order higher than the optimal order for velocity in both the energy norm and the $L^2$-norm and for pressure in the $L^2$-norm are proved in our proposed scheme. The $H$(div)-preserving operator has been constructed based on the polygonal mesh for arbitrary polynomial degrees and employed in the body source assembling to break the locking phenomenon induced by poor mass conservation in the classical discretization. Moreover, the velocity error in our proposed scheme is proved to be independent of pressure and thus we confirm the pressure-robustness. For Stokes simulation, our proposed scheme only modifies the body source assembling but keeps the same stiffness matrix. Four numerical experiments are conducted to validate the convergence results and robustness.

中文翻译：

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多面网格上斯托克斯方程的一种无稳定器、压力稳健和超收敛的弱伽辽金有限元方法

SIAM 科学计算杂志，第 43 卷，第 4 期，第 A2614-A2637 页，2021 年 1 月。
在本文中，我们提出了一种新的无稳定器和压力鲁棒性弱 Galerkin 有限元方法，用于在主要速度-压力公式中的多面网格上具有超收敛的 Stokes 方程。我们提出的方案证明了收敛率比能量范数和 $L^2$-范数中的速度的最佳阶数以及 $L^2$-范数中的压力的​​最佳阶数高一个阶次。$H$(div)-preserving 算子是基于任意多项式次数的多边形网格构建的，并用于体源组装，以打破经典离散化中质量守恒差引起的锁定现象。此外，我们提出的方案中的速度误差被证明与压力无关，因此我们确认了压力稳健性。对于斯托克斯模拟，我们提出的方案仅修改体源组装，但保持相同的刚度矩阵。进行了四次数值实验以验证收敛结果和鲁棒性。