Direct serendipity and mixed finite elements on convex quadrilaterals
Direct serendipity and mixed finite elements on convex quadrilaterals
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DOI：
10.1007/s00211-022-01274-3
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发表时间：
2022-03
影响因子：
2.1
通讯作者：
T. Arbogast;Zhenzhen Tao;Chuning Wang
中科院分区：
文献类型：
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作者：
T. Arbogast;Zhenzhen Tao;Chuning Wang
经典的Serendipity和混合有限元空间在非退化凸四边形上的逼近能力较差。在本文中，我们发展了一族直接偶然性和直接混合有限元空间，它们具有最佳逼近性质和最小的局部维数。用于偶然元素或混合元素的局部形状函数的集合分别包含直接在每个元素上定义的阶的标量或向量多项式的完整集合（即，不从参考元素映射）。由于没有足够的自由度来满足整体或一致性，因此，当满足时，必须向每个单元添加两个补充形函数，当满足时，只需添加一个补充形函数。补充函数的特定选择产生不同的直接元素族。这些新的空间是通过一个德拉姆复杂。对于指数，新的serendipity空间族是我们的直接混合有限元空间在旋度算子下的先驱，它可以被构造成具有约化或全逼近性质。一种直接的偶然性补充剂的选择给出了最近引入的Arbogast-Correa空间的前体（SIAM J Numer Anal 54：3332-3356，2016）。https://doi.org/10.1137/15M1013705).完全直接的偶然性补充可以在不使用来自参考元素的映射的情况下被定义，并且这些又引起完全直接的混合空间。我们的发展是建设性的，因此我们能够为我们的空间提供全球基础。数值结果说明了它们的性质。
The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop families ofdirect serendipityanddirect mixedfinite element spaces, which achieve optimal approximation properties and have minimal local dimension. The set of local shape functions for either the serendipity or mixed elements contains the full set of scalar or vector polynomials of degreer, respectively, defined directly on each element (i.e., not mapped from a reference element). Because there are not enough degrees of freedom for globalorconformity, exactly two supplemental shape functions must be added to each element when, and only one when. The specific choice of supplemental functions gives rise to different families of direct elements. These new spaces are related through a de Rham complex. For index, the new families of serendipity spacesare the precursors under the curl operator of our direct mixed finite element spaces, which can be constructed to have reduced or fullapproximation properties. One choice of direct serendipity supplements gives the precursor of the recently introduced Arbogast–Correa spaces (SIAM J Numer Anal 54:3332–3356, 2016. https://doi.org/10.1137/15M1013705). Otherfullydirect serendipity supplements can be defined without the use of mappings from reference elements, and these give rise in turn tofullydirect mixed spaces. Our development is constructive, so we are able to give global bases for our spaces. Numerical results are presented to illustrate their properties.