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Optimal error estimates of an \(H^1\)-Galerkin mixed finite element method for nonlinear Kirchhoff-type problem

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Abstract

An \(H^1\)-Galerkin mixed finite element method (MFEM) is presented for the nonlinear Kirchhoff-type problem by the bilinear element \(Q_{11}\) and zero order Raviart-Thomas element \(Q_{10}\times Q_{01}\). Optimal error estimates of order O(h) and \(O(h+\tau ^2)\) for the original variable u in \(H^1\) norm and the flux variable \( \vec {p}=\nabla u\) in \(H(\mathrm {\textrm{div}};\Omega )\) norm about the semi-discrete scheme and the linearized fully-discrete scheme are derived, respectively. Finally, some numerical results are provided to verify the theoretical analysis.

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The data that support the findings of this study are available from the corresponding author, upon reasonable request.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China(Nos. 12201640, 12071443).

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Authors and Affiliations

  1. PLA Information Engineering University, Zhengzhou, 450001, China

    Yanmi Wu

  2. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Dongyang Shi

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  1. Yanmi Wu
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Correspondence to Yanmi Wu.

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Communicated by Frederic Valentin.

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Wu, Y., Shi, D. Optimal error estimates of an \(H^1\)-Galerkin mixed finite element method for nonlinear Kirchhoff-type problem. Comp. Appl. Math. 43, 55 (2024). https://doi.org/10.1007/s40314-023-02549-7

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  • DOI: https://doi.org/10.1007/s40314-023-02549-7

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