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This paper concerns with mesh restrictions that are needed to satisfy severalimportant mathematical properties -- maximum principles, comparison principles,and the non-negative constraint -- for a general linear second-order ellipticpartial differential equation. We critically review some recent developments inthe field of discrete maximum principles, derive new results, and discuss somepossible future research directions in this area. In particular, we deriverestrictions for a three-node triangular (T3) element and a four-nodequadrilateral (Q4) element to satisfy comparison principles, maximumprinciples, and the non-negative constraint under the standard single-fieldGalerkin formulation. Analysis is restricted to uniformly elliptic lineardifferential operators in divergence form with Dirichlet boundary conditionsspecified on the entire boundary of the domain. Various versions of maximumprinciples and comparison principles are discussed in both continuous anddiscrete settings. In the literature, it is well-known that an acute-angledtriangle is sufficient to satisfy the discrete weak maximum principle for pureisotropic diffusion. An iterative algorithm is developed to constructsimplicial meshes that preserves discrete maximum principles using existingopen source mesh generators. Various numerical examples based on differenttypes of triangulations are presented to show the pros and cons of placingrestrictions on a computational mesh. We also quantify local and global massconservation errors using representative numerical examples, and illustrate theperformance of metric-based meshes with respect to mass conservation.

On mesh restrictions to satisfy comparison principles, maximum principles, and the non-negative constraint: Recent developments and new results