Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method

IF 0.6 4区数学 Q4 STATISTICS & PROBABILITY

Esaim-Probability and Statistics Pub Date : 2023-01-01 DOI:10.1051/m2an/2022067

Daniele Antonio Di Pietro, Jérôme Droniou

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引用次数: 4

Abstract

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.

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多拓扑配合物的同源性和解析性保持偶然性框架及其在DDR方法中的应用

在这项工作中，我们从广泛的角度研究了通过与希尔伯特配合物兼容的多面体方法的偶然性技术来降低自由度。我们首先建立了一个抽象框架，给定两个由渐变映射连接的配合物，确定了一组能够从一个配合物转移到另一个配合物的同源和分析性质的性质。这个抽象框架的设计考虑了离散的复合体，其中一个是另一个的简化版本，例如在将意外发现技术应用于数值方法时发生的情况。然后，我们使用这个框架作为总体蓝图来设计一个意外的DDR复合体。由于高阶重建和意外发现的结合使用，该复合体在自由度(DOF)计数方面优于之前介绍的所有其他多面体方法，也优于某些元素几何上的有限元。通过两个模型问题(静磁模型和Stokes方程)对自由度减少所产生的增益进行了数值计算。

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来源期刊

Esaim-Probability and Statistics STATISTICS & PROBABILITY-

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original research and survey papers in the area of Probability and Statistics. It covers theoretical and practical aspects, in any field of these domains. Of particular interest are methodological developments with application in other scientific areas, for example Biology and Genetics, Information Theory, Finance, Bioinformatics, Random structures and Random graphs, Econometrics, Physics. Long papers are very welcome. Indeed, we intend to develop the journal in the direction of applications and to open it to various fields where random mathematical modelling is important. In particular we will call (survey) papers in these areas, in order to make the random community aware of important problems of both theoretical and practical interest. We all know that many recent fascinating developments in Probability and Statistics are coming from "the outside" and we think that ESAIM: P&S should be a good entry point for such exchanges. Of course this does not mean that the journal will be only devoted to practical aspects.