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New low-order mixed finite element methods for linear elasticity

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Abstract

New low-order \({H}({{\text {div}}})\)-conforming finite elements for symmetric tensors are constructed in arbitrary dimension. The space of shape functions is defined by enriching the symmetric quadratic polynomial space with the \({(d+1)}\)-order normal-normal face bubble space. The reduced counterpart has only \({d(d+1)}^{{2}}\) degrees of freedom. Basis functions are explicitly given in terms of barycentric coordinates. Low-order conforming finite element elasticity complexes starting from the Bell element, are developed in two dimensions. These finite elements for symmetric tensors are applied to devise robust mixed finite element methods for the linear elasticity problem, which possess the uniform error estimates with respect to the Lamé coefficient \({\lambda }\), and superconvergence for the displacement. Numerical results are provided to verify the theoretical convergence rates.

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Acknowledgements

We thank the editors and anonymous reviewers for their valuable comments and suggestions.

Funding

The first author was supported by the National Natural Science Foundation of China Project 12171300, and the Natural Science Foundation of Shanghai 21ZR1480500.

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  1. Chao Zhang, Yaqian Zhou, and Yangxing Zhu contributed equally to this work

Authors and Affiliations

  1. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Xuehai Huang, Chao Zhang & Yaqian Zhou

  2. Network and Information Technology Center, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Yangxing Zhu

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  1. Xuehai Huang
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Correspondence to Chao Zhang.

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Huang, X., Zhang, C., Zhou, Y. et al. New low-order mixed finite element methods for linear elasticity. Adv Comput Math 50, 17 (2024). https://doi.org/10.1007/s10444-024-10112-z

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