A family of mixed-mixed strongly conservative finite element methods for Biot's model of consolidation
Abstract
We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on H(div div)-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.
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BibTeX
@misc{hong2026family,
title = {{A family of mixed-mixed strongly conservative finite element methods for Biot's model of consolidation}},
author = {Qingguo Hong and Johannes Kraus and Maria Lymbery},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.16418},
eprint = {2607.16418},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.16418},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A family of mixed-mixed strongly conservative finite element methods for Biot's model of consolidation
AU - Qingguo Hong
AU - Johannes Kraus
AU - Maria Lymbery
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2607.16418
UR - https://arxiv.org/abs/2607.16418
AB - We consider a four-field formulation of Biot's quasi-static model of consolidation with the (effective) elastic stress, the solid displacement, the fluid flux and the fluid pressure as unknown physical quantities of interest. The weak form of this system composed of the momentum and mass balance equations complemented by the stress-strain relation and Darcy's law yields a two-fold perturbed saddle-point problem. Its well-posedness is proven for a natural choice of inifinite-dimensional Hilbert spaces, where the displacement-flux pair is sought in a novel Hilbert space. Next, based on recent works on H(div div)-conforming finite elements, we propose a family of mixed-mixed finite element methods that preserve the angular momentum as well as the fluid mass balance pointwise, i.e., in a strong sense. We prove the well-posedness of the arising discrete problem and establish optimal a priori error estimates for the related family of fully conservative mixed-mixed finite element methods.
ER -