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

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.