发表机构
Washington University in St. Louis（圣路易斯华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对有限元外微分中的Hodge-Dirac和Hodge-Laplace问题，提出了任意维数下的HDG方法，采用等阶多项式空间和常数惩罚参数，实现了最优收敛阶，并适用于低正则性和非凸域情形。
AI 中文摘要
我们针对有限元外微分中的两个核心问题——Hodge-Dirac问题和Hodge-Laplace问题，在任意维数$n$下，开发并分析了HDG（混合间断伽辽金）方法。我们的分析允许使用等阶多项式空间和$\u00d7(1)$惩罚参数，这与先前关于Hodge-Laplace问题的工作形成对比，后者需要底层协调复形以及$\u00d7(h)$或$\u00d7(h^{-1})$的惩罚。对于使用$r$次多项式的Hodge-Dirac问题，在适当的正则性假设下，我们的误差估计在所有形式阶数上给出了最优的$(r+1)$阶收敛。对于Hodge-Laplace问题，我们证明了$k$-形式解的最优$(r+1)$阶收敛，以及$(k \u00b1 1)$-形式辅助变量的$(r+\frac{1}{2})$阶收敛。当$k=1$和/或$k=n-1$时，我们获得了改进的$(r+1)$阶辅助变量估计，这特别地改进了最近关于二维向量泊松方程的HDG误差估计。我们的分析还涵盖了较低正则性的情况，包括非凸域上具有凹角奇异性的解，并且仅需要温和的网格正则性条件。结果通过二维和三维的数值实验进行了说明。
英文摘要
We develop and analyze HDG methods for two central problems in finite element exterior calculus, the Hodge-Dirac problem and the Hodge-Laplace problem, in arbitrary dimension $n$. Our analysis allows for equal-order polynomial spaces with $\mathcal{O}(1)$ penalty parameters, by contrast with previous work on the Hodge-Laplace problem requiring an underlying conforming complex and $\mathcal{O}(h)$ or $\mathcal{O}(h^{-1})$ penalties. For the Hodge-Dirac problem with degree-$r$ polynomials, our error estimates give optimal order-$(r+1)$ convergence in all form degrees under suitable regularity hypotheses. For the Hodge-Laplace problem, we prove optimal order-$(r+1)$ convergence for the $k$-form solution and order-$(r+\frac{1}{2})$ convergence for the $(k \pm 1)$-form auxiliary variables. We obtain improved order-$(r+1)$ auxiliary-variable estimates when $k=1$ and/or $k=n-1$, which in particular sharpens some recent HDG error estimates for the two-dimensional vector Poisson equation. Our analysis also encompasses cases of lower regularity, including solutions with reentrant-corner singularities on non-convex domains, and requires only mild mesh regularity conditions. The results are illustrated by numerical experiments in dimensions two and three.
Comments38 pages
