  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.
