On the performance of high-order finite elements with respect to maximum principles and the non-negative constraint for diffusion-type equations

TL;DR

This paper investigates how increasing polynomial order in high-order finite element methods affects maximum principles and non-negativity constraints in anisotropic diffusion problems, revealing limitations and guiding future methodological development.

Contribution

It provides a comprehensive analysis of p-refinement effects on maximum principles and non-negativity in high-order finite elements for diffusion equations, highlighting challenges and future research directions.

Findings

Non-negative constraint violations persist with p-refinement.

High-order methods may not inherently satisfy maximum principles.

Guidance for developing new enforcement methodologies.

Abstract

The main aim of this paper is to document the performance of $p$-refinement with respect to maximum principles and the non-negative constraint. The model problem is (steady-state) anisotropic diffusion with decay (which is a second-order elliptic partial differential equation). We considered the standard single-field formulation (which is based on the Galerkin formalism) and two least-squares-based mixed formulations. We have employed non-uniform Lagrange polynomials for altering the polynomial order in each element, and we have used $p = 1, ..., 10$. It will be shown that the violation of the non-negative constraint will not vanish with $p$-refinement for anisotropic diffusion. We shall illustrate the performance of $p$-refinement using several representative problems. The intended outcome of the paper is twofold. Firstly, this study will caution the users of high-order approximations about its performance with respect to maximum principles and the non-negative constraint. Secondly, this study will help researchers to develop new methodologies for enforcing maximum principles and the non-negative constraint under high-order approximations.