An augmented mixed finite element method for the Biot problem with nonlinear permeability
Abstract
In two and three dimensions, we analyze a mixed formulation of the nonlinear Biot equations in terms of the poroelastic stress tensor, the displacement, the pressure, and the rotation tensor. To establish the well-posedness of the problem, we introduce suitable stabilization terms, leading to an augmented mixed formulation in which the associated parameters are chosen appropriately. Owing to the nonlinearity of the problem, we employ a fixed-point strategy to prove the existence and uniqueness of solutions. We propose a finite element scheme for which we establish existence and uniqueness of a discrete solution, together with a priori error estimates. Additionally, we design and analyze an a posteriori error estimator of the residual type which results to be reliable and efficient. Finally, we present a series of numerical experiments in two and three dimensions in order to assess the performance of the proposed method.
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BibTeX
@misc{lepe2026augmented,
title = {{An augmented mixed finite element method for the Biot problem with nonlinear permeability}},
author = {Felipe Lepe and Gonzalo Rivera and Ricardo Ruiz-Baier and Segundo Villa-Fuentes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37618},
eprint = {2609.37618},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37618},
}
RIS
TY - UNPB
TI - An augmented mixed finite element method for the Biot problem with nonlinear permeability
AU - Felipe Lepe
AU - Gonzalo Rivera
AU - Ricardo Ruiz-Baier
AU - Segundo Villa-Fuentes
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37618
UR - https://arxiv.org/abs/2609.37618
AB - In two and three dimensions, we analyze a mixed formulation of the nonlinear Biot equations in terms of the poroelastic stress tensor, the displacement, the pressure, and the rotation tensor. To establish the well-posedness of the problem, we introduce suitable stabilization terms, leading to an augmented mixed formulation in which the associated parameters are chosen appropriately. Owing to the nonlinearity of the problem, we employ a fixed-point strategy to prove the existence and uniqueness of solutions. We propose a finite element scheme for which we establish existence and uniqueness of a discrete solution, together with a priori error estimates. Additionally, we design and analyze an a posteriori error estimator of the residual type which results to be reliable and efficient. Finally, we present a series of numerical experiments in two and three dimensions in order to assess the performance of the proposed method.
ER -