We provide a unified continuum formulation of linearized mechanics, Stokes' flow and poromechanics in terms of a conservation structure. Starting from this formulation, we construct corresponding simple and robust finite volume discretizations for these physical systems, based only on co-located, cell-centered variables. These discretizations have a minimal discretization stencil, using only the two neighboring cells to a face to calculate numerical stresses and fluxes. We show well-posedness of a weak statement of the continuous formulation in appropriate Hilbert spaces, and identify the appropriate weighted norms for the problem. For the discrete approximations, we prove stability and convergence, both of which are robust in terms of the material parameters. Numerical experiments in 3D support the theoretical results, and provide additional insight into the practical performance of the discretization.

翻译：我们基于守恒结构，为线性化力学、斯托克斯流及多孔力学提供了一个统一的连续介质表述框架。以此表述为出发点，我们针对这些物理系统构建了相应的简单且鲁棒的有限体积离散格式，该格式仅依赖于同位布置的单元中心变量。这些离散格式采用极简的离散模板，仅利用界面两侧相邻的两个单元来计算数值应力与通量。我们证明了连续表述的弱形式在适当希尔伯特空间中的适定性，并确定了该问题所对应的加权范数。对于离散近似，我们证明了其稳定性和收敛性，且两者在材料参数方面均具有鲁棒性。三维数值实验验证了理论结果，并进一步揭示了该离散格式的实际性能。