A nodally bound-preserving finite element method for hyperbolic convection-reaction problems

TL;DR

This paper introduces a finite element method that guarantees physical bounds in hyperbolic convection-reaction problems, ensuring stability and accuracy through a variational inequality framework and extensive numerical validation.

Contribution

The paper develops a nodally bound-preserving finite element approach with rigorous error analysis for hyperbolic problems, a novel framework for ensuring physical bounds at the discrete level.

Findings

Optimal convergence rates demonstrated

Method effectively preserves physical bounds

Prevents unphysical oscillations in nonlinear scenarios

Abstract

In this article, we present a numerical approach to ensure the preservation of physical bounds on the solutions to linear and nonlinear hyperbolic convection-reaction problems at the discrete level. We provide a rigorous framework for error analysis, formulating the discrete problem as a variational inequality and demonstrate optimal convergence rates in a natural norm. We summarise extensive numerical experiments validating the effectiveness of the proposed methods in preserving physical bounds and preventing unphysical oscillations, even in challenging scenarios involving highly nonlinear reaction terms.