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A Rigorous Condition Number Estimate of an Immersed Finite Element Method

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Abstract

It is known that the convergence rate of the traditional iteration methods like the conjugate gradient method depends on the condition number of the stiffness matrix. Moreover the construction of fast solvers like multigrid and domain decomposition methods also need to estimate the condition number of the stiffness matrix. The main purpose of this paper is to give a rigorous condition number estimate of the stiffness matrix resulting from the linear and bilinear immersed finite element approximations of the high-contrast interface problem. It is shown that the condition number is \(C\rho h^{-2}\), where \(\rho \) is the jump of the discontinuous coefficients, h is the mesh size, and the constant C is independent of \(\rho \) and the location of the interface on the triangulation. Numerical results are also given to verify our theoretical findings.

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Acknowledgements

The work of Saihua Wang and Xuejun Xu was supported by National Natural Science Foundation of China (Grant Nos. 11671302, 11871272). Feng Wang was supported by National Natural Science Foundation of China (Grant Nos. 11871281, 11871272). The author would like to thank the editor and the anonymous referees, who meticulously read through the paper and made valuable suggestions and comments which improve this paper greatly.

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Authors and Affiliations

  1. School of Mathematical Science, Tongji University, Shanghai, 200092, China

    Saihua Wang & Xuejun Xu

  2. School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China

    Feng Wang

  3. Institute of Computational Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Xuejun Xu

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  1. Saihua Wang
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  2. Feng Wang
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  3. Xuejun Xu
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Correspondence to Xuejun Xu.

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Wang, S., Wang, F. & Xu, X. A Rigorous Condition Number Estimate of an Immersed Finite Element Method. J Sci Comput 83, 29 (2020). https://doi.org/10.1007/s10915-020-01212-1

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  • DOI: https://doi.org/10.1007/s10915-020-01212-1

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