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A unified analysis of a class of quadratic finite volume element schemes on triangular meshes

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Abstract

This paper presents a general framework for the coercivity analysis of a class of quadratic finite volume element (FVE) schemes on triangular meshes for solving elliptic boundary value problems. This class of schemes covers all the existing quadratic schemes of Lagrange type. With the help of a new mapping from the trial function space to the test function space, we find that each element matrix can be decomposed into three parts: the first part is the element stiffness matrix of the standard quadratic finite element method (FEM), the second part is the difference between the FVE and FEM on the element boundary, while the third part can be expressed as the tensor product of two vectors. Thanks to this decomposition, we obtain a sufficient condition to guarantee the existence, uniqueness, and coercivity result of the FVE solution on triangular meshes. Moreover, based on this sufficient condition, some minimum angle conditions with simple, analytic, and computable expressions can be derived and they depend only on the constructive parameters of the schemes. As a byproduct, some existing coercivity results are improved. Finally, an optimal H1 error estimate is proved by the standard techniques.

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Acknowledgments

The authors would like to thank the reviewers for their careful readings and valuable suggestions.

Funding

This work was partially supported by the National Natural Science Foundation of China (No. 11871009), CAEP Foundation (No. CX2019028), and Guangdong Natural Science Foundation (No. 2017B030311001).

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, People’s Republic of China

    Yanhui Zhou & Jiming Wu

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  1. Yanhui Zhou
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  2. Jiming Wu
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Correspondence to Jiming Wu.

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Communicated by: Aihui Zhou

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Zhou, Y., Wu, J. A unified analysis of a class of quadratic finite volume element schemes on triangular meshes. Adv Comput Math 46, 71 (2020). https://doi.org/10.1007/s10444-020-09809-8

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