Generalization of Strang-Fix remark for discrete maximum principles on tetrahedral meshes
Abstract
Most of qualitative properties of finite element (FE) approximations for various PDE models, in particular the validity of the discrete maximum principles (DMPs), are usually guaranteed by certain geometric properties (e.g. acuteness/nonobtuseness) of FE meshes used. In this work, we revisit the linear FE discretization of the Poisson equation and discuss, in the context of DMPs, the representation of the stiffness matrix entries in terms of the edge lengths and cotangents of the dihedral angles of the tetrahedral meshes, thus avoiding the common use of face areas and element volumes. Using this type of entry representation, certain structural properties of meshes, and simple trigonometric relations, as proposed in the two-dimensional case by Strang and Fix, we show that DMPs still hold (as the stiffness matrices remain M-matrices) for many tetrahedral configurations with obtuse dihedral angles. Such a relaxation of the standard acuteness/nonobtuseness conditions can be vital in practical calculations, as these angle properties are difficult to preserve in global and local mesh refinements. Several representative examples are presented.
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BibTeX
@misc{ali2026generalization,
title = {{Generalization of Strang-Fix remark for discrete maximum principles on tetrahedral meshes}},
author = {Safdar Ali and Maksat Ashyraliyev and Sergey Korotov and Sergei Silvestrov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20403},
eprint = {2610.20403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20403},
}
RIS
TY - UNPB
TI - Generalization of Strang-Fix remark for discrete maximum principles on tetrahedral meshes
AU - Safdar Ali
AU - Maksat Ashyraliyev
AU - Sergey Korotov
AU - Sergei Silvestrov
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20403
UR - https://arxiv.org/abs/2610.20403
AB - Most of qualitative properties of finite element (FE) approximations for various PDE models, in particular the validity of the discrete maximum principles (DMPs), are usually guaranteed by certain geometric properties (e.g. acuteness/nonobtuseness) of FE meshes used. In this work, we revisit the linear FE discretization of the Poisson equation and discuss, in the context of DMPs, the representation of the stiffness matrix entries in terms of the edge lengths and cotangents of the dihedral angles of the tetrahedral meshes, thus avoiding the common use of face areas and element volumes. Using this type of entry representation, certain structural properties of meshes, and simple trigonometric relations, as proposed in the two-dimensional case by Strang and Fix, we show that DMPs still hold (as the stiffness matrices remain M-matrices) for many tetrahedral configurations with obtuse dihedral angles. Such a relaxation of the standard acuteness/nonobtuseness conditions can be vital in practical calculations, as these angle properties are difficult to preserve in global and local mesh refinements. Several representative examples are presented.
ER -