A unified analysis of maximum-norm estimates for a class of HDG methods for parabolic problem in polyhedral domains

Abstract

This paper studies a general class of semi-discrete hybridizable discontinuous Galerkin (HDG) methods, including mixed methods, for parabolic problems in nonconvex polygonal and polyhedral domains. By developing local energy error estimates together with energy estimates for a regularized Green's function, we establish a unified framework to prove the maximum-norm stability of both the semigroup defined by the semi-discrete scheme and the corresponding discrete solutions. The stability analysis shows that the main challenges stem from the treatment of numerical flux variables and the inherent asymmetry of the discrete scheme. These challenges are intrinsic to numerical approaches formulated within the mixed framework for parabolic equations. The asymmetry prevents the direct application of the double kick-back argument, while the presence of flux variables requires special techniques to control their values at the initial time. We emphasize that the analytical tools developed here to address these challenges are sufficiently general to be adapted for maximum-norm stability investigations of a broader class of numerical schemes arising from mixed formulations of parabolic equations. Furthermore, by the stability results and their proofs, we derive the maximal regularity of the semi-discrete solution in $L^{\infty}((0,T);L^p(Ω))$-norm and reduce the maximum-norm error estimates to those of the corresponding elliptic equations and the $L^2$-orthogonal projection. Since the derivation of the local energy error estimates does not rely on the lifting operator, which is only available for simplicial meshes, our results (excluding mixed methods) remain valid for polygonal/polyhedral meshes.