离散微分形式的p版本离散紧致性
Discrete compactness for the p-version of discrete differential forms
AI总结：
本文证明了两维和三维空间中非椭圆变分特征值问题的p版本有限元逼近的离散紧致性，通过引入光滑Poincaré提升算子，为任意阶离散微分形式建立了广义离散紧致性的充分条件，并验证了多种常用边缘有限元族满足该性质。
摘要：
本文证明了两维和三维空间中一类广泛的非椭圆变分特征值问题的p版本有限元逼近的离散紧致性。在一个非常一般的框架下，我们找到了p版本广义离散紧致性的充分条件，该条件是在d维多面体域上任意阶离散微分形式的设定下表述的。分析的主要工具之一是最近引入的光滑Poincaré提升算子[M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]。对于1阶形式，我们的分析表明，几个广泛使用的边缘有限元族在p版本下满足离散紧致性，从而为Maxwell特征值问题提供收敛解。特别地，我们的理论涵盖了三角形和四面体上的Nédélec元（第一类和第二类）以及平行四边形和平行六面体上的Nédélec元（第一类）。
英文摘要：
In this paper we prove the discrete compactness property for a wide class of p-version finite element approximations of non-elliptic variational eigenvalue problems in two and three space dimensions. In a very general framework, we find sufficient conditions for the p-version of a generalized discrete compactness property, which is formulated in the setting of discrete differential forms of any order on a d-dimensional polyhedral domain. One of the main tools for the analysis is a recently introduced smoothed Poincaré lifting operator [M. Costabel and A. McIntosh, On Bogovskii and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Z., (2010)]. For forms of order 1 our analysis shows that several widely used families of edge finite elements satisfy the discrete compactness property in p-version and hence provide convergent solutions to the Maxwell eigenvalue problem. In particular, Nédélec elements on triangles and tetrahedra (first and second kind) and on parallelograms and parallelepipeds (first kind) are covered by our theory.
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