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Coercivity results of a modified Q 1-finite volume element scheme for anisotropic diffusion problems

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Abstract

In this paper, we study a so-called modified Q 1-finite volume element scheme that is obtained by employing the trapezoidal rule to approximate the line integrals in the classical Q 1-finite volume element method. A necessary and sufficient condition is obtained for the positive definiteness of a certain element stiffness matrix. Based on this result, a sufficient condition is suggested to guarantee the coercivity of the scheme on arbitrary convex quadrilateral meshes. When the diffusion tensor is an identity matrix, this sufficient condition reduces to a geometric one, covering some standard meshes, such as the traditional h 1+γ-parallelogram meshes and some trapezoidal meshes. More interesting is that, this sufficient condition has explicit expression, by which one can easily judge on any diffusion tensor and any mesh with any mesh size h > 0. The H 1 error estimate of the modified Q 1-finite volume element scheme is obtained without the traditional h 1+γ-parallelogram assumption. Some numerical experiments are carried out to validate the theoretical analysis.

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Acknowledgments

The authors would like to thank the reviewers for their careful readings and useful suggestions. This work was partially supported by the National Natural Science Foundation of China (Nos. 91330205, 11671313, 11771052) and Foundation of President of China Academy of Engineering Physics (2014-1-042).

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Authors and Affiliations

  1. Graduate School of China Academy of Engineering Physics, Beijing, 100088, China

    Qi Hong

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Jiming Wu

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  1. Qi Hong
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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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Hong, Q., Wu, J. Coercivity results of a modified Q 1-finite volume element scheme for anisotropic diffusion problems. Adv Comput Math 44, 897–922 (2018). https://doi.org/10.1007/s10444-017-9567-3

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  • DOI: https://doi.org/10.1007/s10444-017-9567-3

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