Pressure Monotonicity for a Rotation-Based Biot Discretization with Circumcentric Mass Lumping
Abstract
We establish cellwise pressure monotonicity for a mixed finite element discretization of the rotation-based Biot system. The spatial discretization is based on compatible discrete de Rham complexes, using standard lowest-order elements for rotation, displacement, Darcy flux, and pressure. For the temporal discretization, we employ the backward Euler method. Exactness of the discrete complexes reduces the mechanics contribution to the pressure Schur complement to a scaled pressure mass matrix, yielding a positive diagonal mechanics capacity. For the Darcy part, positive circumcentric facet weights yield a weighted cell graph Laplacian after flux elimination, while the mass-lumped product remains uniformly equivalent to the standard Darcy product. The diagonal mechanics capacity and the weighted cell graph Laplacian make the condensed pressure matrix Stieltjes. Consequently, the pressure propagator is entrywise nonnegative, which yields a cellwise discrete maximum principle. Numerical experiments confirm the predicted convergence rates and show that the proposed discretization suppresses spurious cellwise pressure oscillations.
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BibTeX
@misc{gu2026pressure,
title = {{Pressure Monotonicity for a Rotation-Based Biot Discretization with Circumcentric Mass Lumping}},
author = {Huipeng Gu and Mingchao Cai and Jingzhi Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04535},
eprint = {2610.04535},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04535},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Pressure Monotonicity for a Rotation-Based Biot Discretization with Circumcentric Mass Lumping
AU - Huipeng Gu
AU - Mingchao Cai
AU - Jingzhi Li
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04535
UR - https://arxiv.org/abs/2610.04535
AB - We establish cellwise pressure monotonicity for a mixed finite element discretization of the rotation-based Biot system. The spatial discretization is based on compatible discrete de Rham complexes, using standard lowest-order elements for rotation, displacement, Darcy flux, and pressure. For the temporal discretization, we employ the backward Euler method. Exactness of the discrete complexes reduces the mechanics contribution to the pressure Schur complement to a scaled pressure mass matrix, yielding a positive diagonal mechanics capacity. For the Darcy part, positive circumcentric facet weights yield a weighted cell graph Laplacian after flux elimination, while the mass-lumped product remains uniformly equivalent to the standard Darcy product. The diagonal mechanics capacity and the weighted cell graph Laplacian make the condensed pressure matrix Stieltjes. Consequently, the pressure propagator is entrywise nonnegative, which yields a cellwise discrete maximum principle. Numerical experiments confirm the predicted convergence rates and show that the proposed discretization suppresses spurious cellwise pressure oscillations.
ER -