On positivity preservation of hybrid discontinuous Galerkin methods on hypergraphs

Abstract Hybrid finite element methods, particularly hybridized discontinuous Galerkin methods, are efficient numerical schemes for discretizing the diffusion equation, which encompasses two main physical prin-ciples: mass conservation and positivity preservation. While the former has been extensively analysed 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.