  This paper concerns with mesh restrictions that are needed to satisfy several
important mathematical properties -- maximum principles, comparison principles,
and the non-negative constraint -- for a general linear second-order elliptic
partial differential equation. We critically review some recent developments in
the field of discrete maximum principles, derive new results, and discuss some
possible future research directions in this area. In particular, we derive
restrictions for a three-node triangular (T3) element and a four-node
quadrilateral (Q4) element to satisfy comparison principles, maximum
principles, and the non-negative constraint under the standard single-field
Galerkin formulation. Analysis is restricted to uniformly elliptic linear
differential operators in divergence form with Dirichlet boundary conditions
specified on the entire boundary of the domain. Various versions of maximum
principles and comparison principles are discussed in both continuous and
discrete settings. In the literature, it is well-known that an acute-angled
triangle is sufficient to satisfy the discrete weak maximum principle for pure
isotropic diffusion. An iterative algorithm is developed to construct
simplicial meshes that preserves discrete maximum principles using existing
open source mesh generators. Various numerical examples based on different
types of triangulations are presented to show the pros and cons of placing
restrictions on a computational mesh. We also quantify local and global mass
conservation errors using representative numerical examples, and illustrate the
performance of metric-based meshes with respect to mass conservation.
