In this paper, we propose a new formulation and a suitable finite element method for the steady coupling of viscous flow in deformable porous media using divergence-conforming filtration fluxes. The proposed method is based on the use of parameter-weighted spaces, which allows for a more accurate and robust analysis of the continuous and discrete problems. Furthermore, we conduct a solvability analysis of the proposed method and derive optimal error estimates in appropriate norms. These error estimates are shown to be robust in the case of large Lam\'e parameters and small permeability and storativity coefficients. To illustrate the effectiveness of the proposed method, we provide a few representative numerical examples, including convergence verification, poroelastic channel flow simulation, and test the robustness of block-diagonal preconditioners with respect to model parameters.

翻译：本文针对可变形多孔介质中粘性流动的稳态耦合问题，提出了一种基于散度相容过滤通量的新变分公式及相应的有限元方法。该方法采用参数加权空间，实现了对连续和离散问题更精确且鲁棒的数学分析。进一步地，我们开展了方法的可解性分析，并在适当的范数下推导了最优误差估计。这些误差估计在大Lame参数、小渗透率及储集系数条件下具有鲁棒性。为验证所提方法的有效性，我们提供了若干代表性数值算例，包括收敛性验证、多孔弹性通道流动模拟，以及块对角预条件器关于模型参数的鲁棒性测试。