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A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTN \(H(\textrm{div})\)-elements

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Abstract

A new div FOSLS mixed finite element method is proposed and analyzed for the first-order system of the general second-order elliptic problems in terms of the scalar variable displacement and the vectorial variable flux. The main feature of the proposed method is to apply a local constant element or (bi,tri)linear element \(L^2\) projection to the div term, together with a mesh-dependent div term. The method is coercive and symmetric, allowing any combination of conforming approximations for both scalar and vectorial variables. More importantly, the method is suitable for general nonaffine quadrilateral Raviart–Thomas (RT) and nonaffine hexahedral Raviart–Thomas–Nédélec (RTN) \(H(\textrm{div})\)-elements. For nonaffine quadrilateral RT elements in two dimensions, the proposed method provides optimal approximations for the scalar and vectorial variables for the combination \(Q_m-\textbf{RT}_\ell \) for all \(m\ge 1, \ell \ge s\) where \(s=0\) corresponds to the constant element \(L^2\) projection and \(s=1\) corresponds to the bilinear element \(L^2\) projection; for nonaffine hexahedral RTN elements in three dimensions, it provides optimal approximations for the scalar variable for the combination \(Q_m-\textbf{RTN}_m\) for all \(m\ge 1\) while suboptimal approximations with order m for the vectorial variable. Numerical results are given to confirm the performance and the theoretical results of the new method.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Funding

This work was supported by National Natural Science Foundation of China (11971366, 11571266 and 11661161017 ) and by the Hubei Key Laboratory of Computational Science, Wuhan University, China (2019CFA007).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Huoyuan Duan, Can Wang & Zhijie Du

  2. School of Artificial Intelligence, Jianghan University, Wuhan, 430056, China

    Zhijie Du

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  1. Huoyuan Duan
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  2. Can Wang
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All authors contributed to the study conception and design. All authors read and approved the final manuscript.

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Correspondence to Huoyuan Duan.

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Duan, H., Wang, C. & Du, Z. A Div FOSLS Method Suitable for Quadrilateral RT and Hexahedral RTN \(H(\textrm{div})\)-elements. J Sci Comput 93, 85 (2022). https://doi.org/10.1007/s10915-022-02043-y

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