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A recovery-based linear C0 finite element method for a fourth-order singularly perturbed Monge-Ampère equation

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Abstract

This paper develops a new recovery-based linear C0 finite element method for approximating the weak solution of a fourth-order singularly perturbed Monge-Ampère equation, which is known as the vanishing moment approximation of the Monge-Ampère equation. The proposed method uses a gradient recovery technique to define a discrete Laplacian for a given linear C0 finite element function (offline), the discrete Laplacian is then employed to discretize the biharmonic operator appeared in the equation. It is proved that the proposed C0 linear finite element method has a unique solution using a fixed point argument and the corresponding error estimates are derived in various norms. Numerical experiments are also provided to verify the theoretical error estimates and to demonstrate the efficiency of the proposed recovery-based linear C0 finite element method.

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Funding

The work of the first author was partially supported by the National Natural Science Foundation of China (No. 11871410), the Natural Science Foundation of Fujian Province of China (No. 2018J01004), and the Fundamental Research Funds for the Central Universities (No. 20720180001). The work of the second author was partially supported by the United States National Science Foundation grants DMS-2012414 and DMS-1620168. The work of the third author was partially supported by the National Natural Science Foundation of China (No. 11871092 and U1930402).

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Hongtao Chen

  2. Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Hongtao Chen

  3. Department of Mathematics, The University of Tennessee, Knoxville, TN, 37996, USA

    Xiaobing Feng

  4. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

  5. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Hongtao Chen
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  2. Xiaobing Feng
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  3. Zhimin Zhang
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Corresponding author

Correspondence to Xiaobing Feng.

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Communicated by: Ilaria Perugia

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Chen, H., Feng, X. & Zhang, Z. A recovery-based linear C0 finite element method for a fourth-order singularly perturbed Monge-Ampère equation. Adv Comput Math 47, 21 (2021). https://doi.org/10.1007/s10444-021-09847-w

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  • DOI: https://doi.org/10.1007/s10444-021-09847-w

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