A priori and a posteriori error bounds for the fully mixed FEM formulation of poroelasticity with stress-dependent permeability Artículo académico uri icon

Abstracto

  • Abstract We develop a family of mixed finite element methods for a model of nonlinear poroelasticity where, due 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 linearized 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 Babuška–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 equilibrium 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 two dimensions and three dimensions.

fecha de publicación

  • 2025

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