可杂交间断伽辽金方法的离散庞加莱不等式与迹不等式
Discrete Poincaré and Trace Inequalities for the Hybridizable Discontinuous Galerkin Method
AI总结：
本文以 Crouzeix-Raviart 空间为桥梁，为 HDG 方法推导出离散庞加莱不等式与迹不等式，并据此证明二阶椭圆问题 HDG 格式在低正则性条件下的适定性与鲁棒性。
AI中文摘要：
本文针对可杂交间断伽辽金（HDG）方法推导了离散庞加莱不等式与迹不等式。我们以 Crouzeix-Raviart 空间作为桥梁，将 Brenner 奠基性工作\cite{brenner2003poincare}中的经典离散泛函工具与可杂交有限元空间联系起来，该空间由定义在单元内部和网格骨架上的分片多项式函数组成。这一方法产生了定制化的不等式，为 HDG 离散的稳定性分析奠定了基础。随后，利用所得框架证明了基于 HDG 的二阶椭圆问题数值格式的适定性与鲁棒性，即使在源项和边界数据仅满足极低正则性假设的条件下也成立。
英文摘要：
In this paper, we derive discrete Poincaré and trace inequalities for the hybridizable discontinuous Galerkin (HDG) method. We employ the Crouzeix-Raviart space as a bridge, connecting classical discrete functional tools from Brenner's foundational work \cite{brenner2003poincare} with hybridizable finite element spaces comprised of piecewise polynomial functions defined both within the element interiors and on the mesh skeleton. This approach yields custom-tailored inequalities that underpin the stability analysis of HDG discretizations. The resulting framework is then used to demonstrate the well-posedness and robustness of HDG-based numerical schemes for second-order elliptic problems, even under minimal regularity assumptions on the source term and boundary data.
