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A Recovery Based Linear Finite Element Method For 1D Bi-Harmonic Problems

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Abstract

We analyze a gradient recovery based linear finite element method to solve bi-harmonic equations and the corresponding eigenvalue problems. Our method uses only \(C^0\) element, which avoids complicated construction of \(C^1\) elements and nonconforming elements. Optimal error bounds under various Sobolev norms are established. Moreover, after a post-processing the recovered gradient is superconvergent to the exact one. Some numerical experiments are presented to validate our theoretical findings. As an application, the new method has been also used to solve 1-D fully nonlinear Monge–Ampère equation numerically.

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Acknowledgments

The first author is supported in part by the National Natural Science Foundation of China Under Grants 11301437, the Natural Science Foundation of Fujian Province of China Under Grant 2013J05015, the Fundamental Research Funds for the Central Universities Under Grant 20720150004. The second author is supported in part by the National Natural Science Foundation of China Under Grants 11471031, 91430216 and the US National Science Foundation through Grant DMS-1419040. The third author is partially supported by the National Natural Science Foundation of China through Grants 11571384, 11428103, and the Natural Science Foundation of Guangdong Province (CN) through Grant 2014A030313179.

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, China

    Hongtao Chen

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

    Zhimin Zhang

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

    Zhimin Zhang

  4. School of Data and Computer Science and Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou, 510275, China

    Qingsong Zou

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  1. Hongtao Chen
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  2. Zhimin Zhang
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  3. Qingsong Zou
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Correspondence to Qingsong Zou.

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Chen, H., Zhang, Z. & Zou, Q. A Recovery Based Linear Finite Element Method For 1D Bi-Harmonic Problems. J Sci Comput 68, 375–394 (2016). https://doi.org/10.1007/s10915-015-0141-1

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  • DOI: https://doi.org/10.1007/s10915-015-0141-1

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