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Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems

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Abstract

In this paper, we introduce a locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the conservative flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.

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Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

First author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01005396). Second author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1A2C1003340). Third author is supported by Brain Pool Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (Grant Number NRF-2020H1D3A2A01041079).

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Authors and Affiliations

  1. Department of Mathematics, Kunsan National University, Kunsan, Republic of Korea

    Gwanghyun Jo

  2. Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea

    Do Y. Kwak

  3. Department of Mathematics, Texas State University, San Marcos, USA

    Young-Ju Lee

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  1. Gwanghyun Jo
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  2. Do Y. Kwak
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  3. Young-Ju Lee
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Correspondence to Young-Ju Lee.

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Jo, G., Kwak, D.Y. & Lee, YJ. Locally Conservative Immersed Finite Element Method for Elliptic Interface Problems. J Sci Comput 87, 60 (2021). https://doi.org/10.1007/s10915-021-01476-1

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