The generalized Biot-Brinkman equations describe the displacement, pressures and fluxes in an elastic medium permeated by multiple viscous fluid networks and can be used to study complex poromechanical interactions in geophysics, biophysics and other engineering sciences. These equations extend on the Biot and multiple-network poroelasticity equations on the one hand and Brinkman flow models on the other hand, and as such embody a range of singular perturbation problems in realistic parameter regimes. In this paper, we introduce, theoretically analyze and numerically investigate a class of three-field finite element formulations of the generalized Biot-Brinkman equations. By introducing appropriate norms, we demonstrate that the proposed finite element discretization, as well as an associated preconditioning strategy, is robust with respect to the relevant parameter regimes. The theoretical analysis is complemented by numerical examples.

翻译：通用的Biot-Brinkman方程式描述了由多粘液流网络渗透的弹性介质中的迁移、压力和通量,可用于研究地球物理学、生物物理学和其他工程科学中复杂的机械相互作用。这些方程式一方面延伸到Biot和多网络孔径度方程式,另一方面延伸到Brinkman流程模型,在现实的参数系统中体现了一系列奇特的扰动问题。在本文中,我们引入、理论上分析和从数字上调查了通用Biot-Brinkman方程式的三类三维有限元素配方。通过引入适当的规范,我们证明拟议的有限元素分解以及相关的前提条件战略在有关参数系统中是健全的。理论分析得到了数字实例的补充。