多面体网格上混合高阶微分形式的一致离散庞加莱不等式
Uniform discrete Poincaré inequalities for Hybrid High-Order differential forms on polyhedral meshes
- IMAG, Univ Montpellier, CNRS（蒙彼利埃大学数学与计算机科学研究室、法国国家科学研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对多面体网格提出混合高阶微分形式框架，证明各形式次数下的一致离散庞加莱不等式，为任意空间维度的外微积分提供统一推广。
AI中文摘要：
我们针对一般多面体网格上的微分形式引入了混合高阶（Hybrid High-Order, HHO）框架。对每个形式次数 $k\in\{0,\ldots,n-1\}$，该离散空间结合了网格单元上的多项式微分形式与网格面上的多项式迹未知量，且外导数通过局部斯托克斯公式重构。在三维空间中，对应的向量代理量可恢复常规的混合梯度、旋度与散度重构，而微分形式框架则提供了任意空间维度下统一的外微积分推广。在该框架内，我们证明了每个形式次数下的一致离散庞加莱不等式：标准的间断-稳定半范数可在混合 $L^2$ 范数意义下，一致控制到给定重构外导数的全核的距离，且该一致性与网格尺寸及面空间的所有稳定性允许选择无关。该结果适用于任意拓扑的区域。
英文摘要：
We introduce a Hybrid High-Order framework for differential forms on general polyhedral meshes and characterise stability-admissible face spaces in terms of the traces of the polynomial kernel of the exterior derivative. The analysis is based on a cellular-to-hybrid transfer mechanism in which the global part of the stability problem is reduced to a single cellular cochain Poincaré problem, and the resulting control is propagated to the hybrid level using stable polynomial skeletons and local polynomial completions. This mechanism yields a uniform stable conforming lifting and a uniform discrete Poincaré inequality for every form degree in arbitrary space dimension. The lifting preserves the prescribed reconstructed exterior derivative after projection, whereas the Poincaré inequality controls the broken cell polynomial form modulo the continuous conforming kernel of the exterior derivative. Both results hold on domains with arbitrary topology and uniformly over stability-admissible choices of the face spaces. In three dimensions, the conforming-kernel estimate recovers the hybrid Poincaré--Wirtinger inequality for the gradient, the second hybrid Weber estimate under the standard gauge, and the corresponding divergence-kernel estimate.
