This paper presents a pressure-robust enriched Galerkin (EG) method for the Brinkman equations with minimal degrees of freedom based on EG velocity and pressure spaces. The velocity space consists of linear Lagrange polynomials enriched by a discontinuous, piecewise linear, and mean-zero vector function per element, while piecewise constant functions approximate the pressure. We derive, analyze, and compare two EG methods in this paper: standard and robust methods. The standard method requires a mesh size to be less than a viscous parameter to produce stable and accurate velocity solutions, which is impractical in the Darcy regime. Therefore, we propose the pressure-robust method by utilizing a velocity reconstruction operator and replacing EG velocity functions with a reconstructed velocity. The robust method yields error estimates independent of a pressure term and shows uniform performance from the Stokes to Darcy regimes, preserving minimal degrees of freedom. We prove well-posedness and error estimates for both the standard and robust EG methods. We finally confirm theoretical results through numerical experiments with two- and three-dimensional examples and compare the methods' performance to support the need for the robust method.

翻译：本文提出了一种基于最小自由度的压力鲁棒富集Galerkin(EG)方法，用于求解Brinkman方程。该方法采用EG速度与压力空间，其中速度空间由线性拉格朗日多项式富集每个单元上的不连续、分片线性且均值为零的向量函数构成，而压力则通过分片常数函数近似。本文推导、分析并比较了两种EG方法：标准方法与鲁棒方法。标准方法要求网格尺寸小于粘性参数才能获得稳定且精确的速度解，这在达西流区域中不切实际。为此，我们提出压力鲁棒方法，通过引入速度重构算子，将EG速度函数替换为重构速度。鲁棒方法可导出与压力项无关的误差估计，并在从斯托克斯流到达西流的全域范围内保持一致性能，同时保留最小自由度。我们证明了标准EG方法与鲁棒EG方法的适定性与误差估计。最后，通过二维与三维数值实验验证理论结果，并对比两种方法的性能以论证鲁棒方法的必要性。