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A quadratic finite volume method for nonlinear elliptic problems

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Abstract

In this paper, we consider a quadratic finite volume method (FVM) for solving second-order nonlinear elliptic problems. Under reasonable assumptions, we shall establish the existence and uniqueness of the quadratic FVM approximation and develop the error analysis of the approximation solution. To be specific, without any additional requirements on the underlying triangular meshes, we derive the optimal error estimate by assumption that \(u\in H^{3}\cap W^{2,\infty }\), where u is the solution of the nonlinear elliptic problems. Numerical experiments are presented to confirm the theoretical results.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions leading to an improvement of this paper.

Funding

This work was supported in part by the National Natural Science Foundation of China under grants 11771375, 11571115, 11901506, and 11571297, and by the Shandong Province Natural Science Foundation under grants ZR2018QA003 and ZR2018MA008.

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, People’s Republic of China

    Yuanyuan Zhang, Chuanjun Chen & Chunjia Bi

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  1. Yuanyuan Zhang
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  2. Chuanjun Chen
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  3. Chunjia Bi
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Correspondence to Chunjia Bi.

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Communicated by: Long Chen

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Zhang, Y., Chen, C. & Bi, C. A quadratic finite volume method for nonlinear elliptic problems. Adv Comput Math 47, 32 (2021). https://doi.org/10.1007/s10444-021-09853-y

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