广义Gaffney不等式与离散紧致性对于离散微分形式
Generalized Gaffney inequality and discrete compactness for discrete differential forms
AI总结：
本文研究了广义Gaffney不等式和离散紧致性，针对离散微分形式在s-正则域上的有限元应用，提出新的Hodge映射及其逼近性质，用于证明若干有限元近似在标量和向量拉普拉斯问题中的L^p估计。
摘要：
我们证明了广义Gaffney不等式和有限元微分形式在s-正则域上的离散紧致性，包括一般Lipschitz域。在计算电磁学中，这些结果的特例已被用于边缘元与弱施加无散条件的分析，以及非线性和特征值问题。本文将这些结果推广到离散微分形式，不必要强施加或弱施加约束。分析依赖于新的Hodge映射及其逼近性质。作为应用，我们展示了若干有限元近似在标量和向量拉普拉斯问题中的L^p估计。
英文摘要：
We prove generalized Gaffney inequalities and the discrete compactness for finite element differential forms on $s$-regular domains, including general Lipschitz domains. In computational electromagnetism, special cases of these results have been established for edge elements with weakly imposed divergence-free conditions and used in the analysis of nonlinear and eigenvalue problems. In this paper, we generalize these results to discrete differential forms, not necessarily with strongly or weakly imposed constraints. The analysis relies on a new Hodge mapping and its approximation property. As an application, we show $L^{p}$ estimates for several finite element approximations of the scalar and vector Laplacian problems.
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