A Note on why Enforcing Discrete Maximum Principles by a simple a Posteriori Cutoff is a Good Idea

TL;DR

This paper demonstrates that a simple a posteriori cutoff effectively enforces discrete maximum principles in PDE approximations, often improving accuracy without geometric restrictions across various finite element methods.

Contribution

It introduces a straightforward cutoff method to enforce maximum principles, applicable to multiple approximation types, enhancing solution quality without geometric constraints.

Findings

The cutoff enforces discrete maximum principles effectively.

It can improve approximation accuracy in the energy norm.

Applicable to conforming higher order and hp-finite elements.

Abstract

Discrete maximum principles in the approximation of partial differential equations are crucial for the preservation of qualitative properties of physical models. In this work we enforce the discrete maximum principle by performing a simple cutoff. We show that for many problems this a posteriori procedure even improves the approximation in the natural energy norm. The results apply to many different kinds of approximations including conforming higher order and $hp$-finite elements. Moreover in the case of finite element approximations there is no geometrical restriction on the partition of the domain.