In this paper, we present a pressure-robust enriched Galerkin (EG) scheme for solving the Stokes equations, which is an enhanced version of the EG scheme for the Stokes problem proposed in [Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]. The pressure-robustness is achieved by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. An a priori error analysis proves that the velocity error is independent of the pressure and viscosity. We also propose and analyze a perturbed version of our pressure-robust EG method that allows for the elimination of the degrees of freedom corresponding to the discontinuous component of the velocity vector via static condensation. The resulting method can be viewed as a stabilized $H^1$-conforming $\mathbb{P}_1$-$\mathbb{P}_0$ method. Further, we consider efficient block preconditioners whose performances are independent of the viscosity. The theoretical results are confirmed through various numerical experiments in two and three dimensions.

翻译：本文提出了一种用于求解Stokes方程的压力鲁棒型富集Galerkin（EG）格式，该格式是对文献[Son-Young Yi, Xiaozhe Hu, Sanghyun Lee, James H. Adler, An enriched Galerkin method for the Stokes equations, Computers and Mathematics with Applications, accepted, 2022]中提出的Stokes问题EG格式的增强版本。通过在离散系统右端载荷向量上引入速度重构算子，实现了压力鲁棒性。先验误差分析表明，速度误差与压力和黏度无关。本文还提出并分析了一种压力鲁棒型EG方法的扰动版本，该版本允许通过静态凝聚消除速度矢量中不连续分量对应的自由度。由此得到的方法可视为具有稳定性的$H^1$协调$\mathbb{P}_1$-$\mathbb{P}_0$方法。此外，我们考虑了性能与黏度无关的高效块预处理子。二维和三维空间中的大量数值实验验证了理论结果。