AI 中文总结
本文针对多面体区域抛物问题的一类HDG方法，建立统一框架证明其最大范数稳定性，推导半离散解的最大正则性，所得结果适用于多边形/多面体网格。
AI 中文摘要
本文研究了一类半离散可杂交间断伽略金（HDG）方法，包括混合方法，用于非凸多边形和多面体区域中的抛物问题。通过开发局部能量误差估计以及正则化格林函数的能量估计，我们建立了一个统一框架，以证明半离散格式定义的半群以及相应离散解的最大范数稳定性。稳定性分析表明，主要挑战源于数值通量变量的处理以及离散格式固有的不对称性，这些挑战是抛物方程混合框架内数值方法所固有的。这种不对称性阻碍了直接应用双重回退论证，而通量变量的存在则需要特殊技术来控制其初始时刻的值。我们强调，本文为应对这些挑战而开发的分析工具具有足够的通用性，可适用于更广泛的抛物方程混合格式产生的数值格式的最大范数稳定性研究。此外，通过稳定性结果及其证明，我们推导了半离散解在$L^{\infty}((0,T);L^p(\Omega))$范数下的最大正则性，并将最大范数误差估计简化为对应椭圆方程的误差估计和$L^2$-正交投影的误差估计。由于局部能量误差估计的推导不依赖仅适用于单纯形网格的提升算子，因此本文的结果（不包括混合方法）对多边形/多面体网格仍然有效。
英文摘要
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.
