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Generalized finite element systems for smooth differential forms and Stokes’ problem

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Abstract

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of finite element systems, and the examples include conforming mixed finite elements for Stokes’ equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.

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Acknowledgements

We are grateful to Richard Falk for pointing out the paper [3], which has interesting connections with this one. We are also grateful to Shangyou Zhang for numerous bibliographical remarks. SHC is supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, Project 278011 STUCCOFIELDS. KH is supported by the China Scholarship Council (CSC), Project 201506010013 and by the European Research Council through the FP7-IDEAS-ERC Advanced Grant scheme, Project 650138 FEEC-A. The stimulating collaborations are achieved during his visit at University of Oslo (UiO) since September 2015. He is grateful for the kind hospitality and support of UiO.

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  1. Department of Mathematics, University of Oslo, PO Box 1053, Blindern, 0316, Oslo, Norway

    Snorre H. Christiansen & Kaibo Hu

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  1. Snorre H. Christiansen
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  2. Kaibo Hu
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Correspondence to Snorre H. Christiansen.

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Christiansen, S.H., Hu, K. Generalized finite element systems for smooth differential forms and Stokes’ problem. Numer. Math. 140, 327–371 (2018). https://doi.org/10.1007/s00211-018-0970-6

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