The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a point-wise divergence-free approximate velocity on cells. However, the approximate velocity is not H(div)-conforming and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. (SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314). We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods.

中文翻译：

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重构算子求解斯托克斯问题的压力鲁棒嵌入式不连续伽辽金方法

用于斯托克斯问题的嵌入式不连续伽辽金 (EDG) 有限元方法导致单元上的逐点无发散近似速度。然而，近似速度不符合 H(div)，可以证明这是 EDG 方法不是压力稳健的原因，即速度的误差取决于连续压力。在本文中，我们提出了一个局部重建算子，它将离散的无发散测试函数映射到完全无发散的测试函数。这种局部重建算子仅通过改变离散化的右侧来恢复压力稳健性，类似于最近由 Lederer 等人为 Taylor-Hood 和 mini 元素引入的重建算子。(SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314)。我们提出了离散化的先验误差分析，显示了速度误差的最佳收敛速度和压力鲁棒性。这些结果由数值例子验证。这项研究的动机是由此产生的 EDG 方法结合了不连续 Galerkin 方法的多功能性、连续 Galerkin 方法的计算效率和压力稳健有限元方法的准确性。