混合维与等维建模的抽象逼近工具
An abstract approximation tool for mixed-dimensional and equidimensional modeling
AI总结：
本文构造重叠开覆盖和有界上链映射，建立单纯形de Rham复形与Čech-de Rham复形子复形间的同构，作为比较混合维与等维问题的抽象逼近工具。
摘要：
工程和科学中的许多耦合问题可以用相邻域上的椭圆型偏微分方程来描述，其中耦合既可以视为模型域之间的薄等维重叠，也可以视为低维界面。由此，我们区分同一系统的等维模型和混合维模型，并且这些建模方法之间的关系具有自然的研究意义。在本文中，我们为一类单纯形几何构造了一个重叠开覆盖，并构造了一个从单纯形de Rham复形到与该重叠覆盖相关联的Čech-de Rham复形的有界上链映射。因此，我们在单纯形de Rham复形（即混合维划分上的函数和形式及其微分）与Čech-de Rham复形的子复形（即等维划分上的函数和形式及其微分）之间建立了同构，这作为比较混合维问题与同一问题的等维版本的抽象逼近工具。
英文摘要：
Many coupled problems in engineering and science can be described by elliptic partial differential equations on adjacent domains, where the coupling can be considered either as a thin equidimensional overlap between the model domains, or as a lower-dimensional interface. Thereby we distinguish equidimensional and mixed-dimensional models of the same system, and the relationship between these modeling approaches is of natural interest. In this paper, we construct an overlapping open cover for a class of simplicial geometries and construct a bounded cochain map from the simplicial de Rham complex to the Čech-de Rham complex associated with the overlapping cover. Thus, we establish an isomorphism between simplicial de Rham complexes (i.e. functions and forms on mixed-dimensional partitions and their differentials) and subcomplexes of Čech-de Rham complexes (i.e. functions and forms on equidimensional partitions and their differentials), which serves as an abstract approximation tool for comparing mixed-dimensional problems to the equidimensional version of the same problem.
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