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

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 $\mathcal{O}(\mathrm{\parallel }\text{V}b{\mathrm{\parallel }}_{\mathrm{\infty }}h+\mathrm{\parallel }c{\mathrm{\parallel }}_{\mathrm{\infty }}{h}^2)$-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 $\text{V}b$ 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 $\mathcal{O}(\mathrm{\parallel }\text{V}b{\mathrm{\parallel }}_{\mathrm{\infty }}h+\mathrm{\parallel }c{\mathrm{\parallel }}_{\mathrm{\infty }}{h}^2)$ 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.