A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability
<|reference_start|>A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability: 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.<|reference_end|>
@article{khan2024a,
    title={A priori and a posteriori error bounds for the fully mixed FEM
  formulation of poroelasticity with stress-dependent permeability},
    author={Arbaz Khan, Bishnu P. Lamichhane, Ricardo Ruiz-Baier, Segundo
  Villa-Fuentes},
    journal={arXiv preprint arXiv:2409.03246},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.03246},
    primaryClass={math.NA cs.NA}
}