论文标题
用于多面体复合物的同源和分析性的串行框架，并应用于DDR方法
Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method
论文作者
论文摘要
在这项工作中，我们从广义的角度研究了与希尔伯特综合体兼容的多种方法的偶然性技术的降低。我们首先建立了一个抽象框架，鉴于两个通过分级图连接的复合物，它标识了一组属性，从而使同源性和分析性能从一个复合物转移到另一个复合物。该抽象框架的设计是具有离散复合物的，其中一个是另一个的简化版本，例如在将偶然性技术应用于数值方法时发生。然后，我们将此框架用作总体蓝图来设计偶然性DDR复合体。得益于高阶重建和偶然性的综合使用，该复合物在自由度（DOF）计数方面与先前引入的所有其他多面有方法相比，对某些元素几何形状的有限元素进行了比较。在两个模型问题上，对DOF数量减少产生的增益进行了数值评估：磁静态模型和Stokes方程。
In this work we investigate from a broad perspective the reduction of degrees of freedom through serendipity techniques for polytopal methods compatible with Hilbert complexes. We first establish an abstract framework that, given two complexes connected by graded maps, identifies a set of properties enabling the transfer of the homological and analytical properties from one complex to the other. This abstract framework is designed having in mind discrete complexes, with one of them being a reduced version of the other, such as occurring when applying serendipity techniques to numerical methods. We then use this framework as an overarching blueprint to design a serendipity DDR complex. Thanks to the combined use of higher-order reconstructions and serendipity, this complex compares favorably in terms of degrees of freedom (DOF) count to all the other polytopal methods previously introduced and also to finite elements on certain element geometries. The gain resulting from such a reduction in the number of DOFs is numerically evaluated on two model problems: a magnetostatic model, and the Stokes equations.
