On positivity preservation of hybrid discontinuous Galerkin methods on hypergraphs

TL;DR

This paper analyzes the positivity preservation of hybridized discontinuous Galerkin methods for diffusion equations, providing theoretical guarantees and counterexamples, supported by numerical experiments.

Contribution

It presents a theorem ensuring positivity of solutions in HDG methods and highlights conditions leading to nonpositive discretizations.

Findings

Theorem guarantees positivity under certain conditions.

Counterexamples show when positivity is not preserved.

Numerical experiments confirm theoretical results.

Abstract

Hybrid finite element methods, particularly hybridized discontinuous Galerkin (HDG) methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical principles: mass conservation and positivity preservation. While the former has been extensively analyzed in the literature, this paper investigates the latter. We state a theorem that guarantees the positivity of both the bulk and skeleton approximations to the primary unknown (concentration) and provide counterexamples for nonpositive discretizations. The theoretical findings are confirmed by numerical experiments.