Maximum principle in linear finite element approximations of anisotropic diffusion-convection-reaction problems

TL;DR

This paper establishes mesh conditions under which linear finite element methods for anisotropic diffusion-convection-reaction problems satisfy a discrete maximum principle, including new weaker conditions in 2D and numerical verification.

Contribution

It introduces new mesh conditions ensuring the maximum principle in finite element approximations for anisotropic problems, generalizing and unifying existing conditions.

Findings

Derived mesh conditions for maximum principle satisfaction.

In 2D, weaker mesh conditions are established.

Numerical results confirm theoretical predictions.

Abstract

A mesh condition is developed for linear finite element approximations of anisotropic diffusion-convection-reaction problems to satisfy a discrete maximum principle. Loosely speaking, the condition requires that the mesh be simplicial and $\mathcal{O}(\|\V{b}\|_\infty h + \|c\|_\infty h^2)$-nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix, where $h$ denotes the mesh size and $\V{b}$ and $c$ are the coefficients of the convection and reaction terms. In two dimensions, the condition can be replaced by a weaker mesh condition (an $\mathcal{O}(\|\V{b}\|_\infty h + \|c\|_\infty h^2)$ perturbation of a generalized Delaunay condition). These results include many existing mesh conditions as special cases. Numerical results are presented to verify the theoretical findings.