This paper presents both a priori and a posteriori error analyses for a really pressure-robust virtual element method to approximate the incompressible Brinkman problem. We construct a divergence-preserving reconstruction operator using the Raviart-Thomas element for the discretization on the right-hand side. The optimal priori error estimates are carried out, which imply the velocity error in the energy norm is independent of both the continuous pressure and the viscosity. Taking advantage of the virtual element method's ability to handle more general polygonal meshes, we implement effective mesh refinement strategies and develop a residual-type a posteriori error estimator. This estimator is proven to provide global upper and local lower bounds for the discretization error. Finally, some numerical experiments demonstrate the robustness, accuracy, reliability and efficiency of the method.

翻译：本文针对不可压缩Brinkman问题的真压力鲁棒虚拟元法，同时给出了先验与后验误差分析。我们利用Raviart-Thomas元构造了保持散度特性的重构算子，用于右端项的离散化处理。推导出的最优先验误差估计表明，能量范数下的速度误差与连续压力和黏度参数均无关。借助虚拟元方法处理更一般多边形网格的能力，我们实施了有效的网格细化策略，并建立了残差型后验误差估计子。该估计子被证明能够为离散误差提供全局上界和局部下界。最后，通过数值实验验证了该方法在鲁棒性、精确性、可靠性和计算效率方面的优越性能。