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Nonstandard finite element de Rham complexes on cubical meshes

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Abstract

Two general operations are proposed on finite element differential complexes on cubical meshes that can be used to construct and analyze sequences of “nonstandard” convergent methods. The first operation, called DoF-transfer, moves edge degrees of freedom to vertices in a way that reduces global degrees of freedom while increasing continuity order at vertices. The second operation, called serendipity, eliminates interior bubble functions and degrees of freedom locally on each element without affecting edge degrees of freedom. These operations can be used independently or in tandem to create nonstandard complexes that incorporate Hermite, Adini and “trimmed-Adini” elements. The resulting elements lead to convergent non-conforming methods for problems requiring stronger regularity and satisfy a discrete Korn inequality. Potential benefits of applying these elements to Stokes, biharmonic and elasticity problems are discussed.

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Authors and Affiliations

  1. Department of Mathematics, University of Arizona, 617 N. Santa Rita Ave., Tucson, AZ, USA

    Andrew Gillette

  2. School of Mathematics, University of Minnesota, 206 Church St. SE, Minneapolis, MN, USA

    Kaibo Hu

  3. LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, and National Centre for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China

    Shuo Zhang

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  1. Andrew Gillette
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  2. Kaibo Hu
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  3. Shuo Zhang
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Correspondence to Kaibo Hu.

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AG was supported in part by National Science Foundation Award DMS-1522289. The research of KH leading to the results of this paper was partly carried out during his affiliation with the University of Oslo, supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement 339643. SZ was supported in part by the National Natural Science Foundation of China with Grants No. 11471026, No. 11871465.

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Gillette, A., Hu, K. & Zhang, S. Nonstandard finite element de Rham complexes on cubical meshes. Bit Numer Math 60, 373–409 (2020). https://doi.org/10.1007/s10543-019-00779-y

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