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Uniformly Convergent Cubic Nonconforming Element For Darcy–Stokes Problem

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Abstract

In this paper we construct a cubic element named DSC33 for the Darcy–Stokes problem of three-dimensional space. The finite element space \({{\varvec{V}}}_{h}\) for velocity is -conforming, i.e., the normal component of a function in \({{\varvec{V}}}_{h}\) is continuous across the element boundaries, meanwhile the tangential component of a function in \({{\varvec{V}}}_{h}\) is averagely continuous across the element boundaries, hence \({{\varvec{V}}}_{h}\) is \({{\varvec{H}}}^{1}\)-average conforming. We prove that this element is uniformly convergent with respect to the perturbation constant \(\varepsilon \) for the Darcy–Stokes problem. In addition, we construct a discrete de Rham complex corresponding to DSC33 element. The finite element spaces in the discrete de Rham complex can be applied to some singular perturbation problems.

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Acknowledgements

We should like to thank the anonymous referees for their helpful suggestions on this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Shao-chun Chen & Ji-kun Zhao

  2. School of Mathematics and Statistics, Institute of Henan Education, Zhengzhou, 450046, China

    Li-na Dong

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  1. Shao-chun Chen
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  2. Li-na Dong
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  3. Ji-kun Zhao
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Correspondence to Li-na Dong.

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This work is supported by NSFC (11371331).

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Chen, Sc., Dong, Ln. & Zhao, Jk. Uniformly Convergent Cubic Nonconforming Element For Darcy–Stokes Problem. J Sci Comput 72, 231–251 (2017). https://doi.org/10.1007/s10915-016-0353-z

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  • DOI: https://doi.org/10.1007/s10915-016-0353-z

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