论文标题
伴随运算符和不合格有限元外观表积分的部分伴随离散
Partially adjoint discretizations of adjoint operators and nonconforming finite element exterior calculus
论文作者
论文摘要
本文涉及希尔伯特空间之间的一对伴随操作员的离散化，以便可以保留伴随性能。由于离散运算符的有限维质本质，在本文中有动机和呈现的新框架，即部分伴随的操作员的理论，因此可以在其他背景空间中弄清楚无需在其他背景空间中定义的有限维操作员的伴随性能。在框架下提出了一种形式的方法，可以通过符合离散化（CD）和每个操作员的伴随的离散化（ABCD）来部分构造离散化。此外，该方法会导致无限维操作员家族的渐近均匀性。理论框架的有效性和离散化的正式构建由伴随外部差异操作员的系统内置离散剂的系统性介绍。 本文中有关的伴随特性是封闭范围定理和强二元性，其预防尚未得到很好的研究。封闭范围定理的量化版本都是为伴随运营商和部分伴随离散化的。概念poincare-alexander-lefschetz（简短的P-A-L）型二元性是用于操作者理论的，并且为伴随的操作员确定了水平和垂直的P-A-L二元性，并确定其类似物以部分相邻离散化。特别是由于外部差异操作员的部分伴随离散化，庞加莱 - lefschetz二元性被保留为以前尚未获得的身份。
This paper concerns the discretizations in pair of adjoint operators between Hilbert spaces so that the adjoint properties can be preserved. Due to the finite-dimensional essence of discretized operators, a new framework, theory of partially adjoint operators, is motivated and presented in this paper, so that adjoint properties can be figured out for finite-dimensional operators which can not be non-trivially densely defined in other background spaces. A formal methodology is presented under the framework to construct partially adjoint discretizations by a conforming discretization (CD) and an accompanied-by-conforming discretization (ABCD) for each of the operators. Moreover, the methodology leads to an asymptotic uniformity of an infinite family of finite-dimensional operators. The validities of the theoretical framework and the formal construction of discretizations are illustrated by a systematic family of in-pair discretizations of the adjoint exterior differential operators. The adjoint properties concerned in the paper are the closed range theorem and the strong dualities, whose preservations have not been well studied yet. Quantified versions of the closed range theorem are established for both adjoint operators and partially adjoint discretizations. The notion Poincare-Alexander-Lefschetz (P-A-L for short) type duality is borrowed for operator theory, and horizontal and vertical P-A-L dualities are figured out for adjoint operators and their analogues are established for partially adjoint discretizations. Particularly by partially adjoint discretizations of exterior differential operators, the Poincare-Lefschetz duality is preserved as an identity, which was not yet obtained before.
