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

TL;DR

This paper reviews recent developments and introduces new results on mesh restrictions necessary to ensure maximum principles, comparison principles, and non-negativity constraints in finite element methods for elliptic PDEs, including mesh construction algorithms.

Contribution

It derives specific mesh restrictions for T3 and Q4 elements to satisfy key mathematical principles and develops an iterative algorithm for mesh generation that preserves these properties.

Findings

Triangular meshes with acute angles satisfy discrete maximum principles.

Mesh restrictions impact mass conservation and solution accuracy.

The proposed mesh construction algorithm effectively preserves maximum principles.

Abstract

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.