We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, thanks to a rewriting of the constitutive equations, the permeability depends on the total poroelastic stress and on the fluid pressure and therefore we can use the Hellinger--Reissner principle with weakly imposed stress symmetry for Biot's equations. The problem is adequately structured into a coupled system consisting of one saddle-point formulation, one linearised perturbed saddle-point formulation, and two off-diagonal perturbations. This system's unique solvability requires assumptions on regularity and Lipschitz continuity of the inverse permeability, and the analysis follows fixed-point arguments and the Babu\v{s}ka--Brezzi theory. The discrete problem is shown uniquely solvable by applying similar fixed-point and saddle-point techniques as for the continuous case. The method is based on the classical PEERS$_k$ elements, it is exactly momentum and mass conservative, and it is robust with respect to the nearly incompressible as well as vanishing storativity limits. We derive a priori error estimates, we also propose fully computable residual-based a posteriori error indicators, and show that they are reliable and efficient with respect to the natural norms, and robust in the limit of near incompressibility. These a posteriori error estimates are used to drive adaptive mesh refinement. The theoretical analysis is supported and illustrated by several numerical examples in 2D and 3D.

翻译：我们针对非线性孔隙弹性力学模型发展了一族混合有限元方法。该模型通过重写本构方程，使得渗透率依赖于总孔隙弹性应力与流体压力，从而能够应用Hellinger-Reissner原理对Biot方程进行弱对称应力约束。该问题被合理构建为由一个鞍点形式、一个线性化扰动鞍点形式以及两个非对角扰动组成的耦合系统。系统唯一可解性需要逆渗透率满足正则性与Lipschitz连续性假设，分析过程采用不动点论证与Babuška-Brezzi理论。通过应用与连续情形类似的不动点及鞍点技术，证明了离散问题的唯一可解性。该方法基于经典PEERS$_k$单元，严格满足动量与质量守恒，并对近不可压缩与趋近零储水系数极限具有鲁棒性。我们推导了先验误差估计，同时提出完全可计算的基于残差的后验误差指示子，证明其在自然范数下具有可靠性与有效性，并在近不可压缩极限下保持鲁棒性。这些后验误差估计被用于驱动自适应网格加密。理论分析通过二维与三维数值算例得到验证与展示。