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EXtended HDG Methods for Second Order Elliptic Interface Problems

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Abstract

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise \(C^2\) smooth interface. It uses piecewise polynomials of degrees k\((k\ge 1)\) and \(k-1\) respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree k for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree \( k-1\) for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in \(L^2\) norm and for the potential approximation in piecewise \(H^1\) seminorm without requiring “sufficiently large” stabilization parameters in the schemes. In addition, error estimation for the potential approximation in \(L^2\) norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.

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Authors and Affiliations

  1. South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, 510630, China

    Yihui Han

  2. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modelling and High Performance Scientific Computing, Xiamen University, Xiamen, 361005, Fujian, China

    Huangxin Chen

  3. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Xiao-Ping Wang

  4. School of Mathematics, Sichuan University, Chengdu, 610064, China

    Xiaoping Xie

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  1. Yihui Han
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  2. Huangxin Chen
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  4. Xiaoping Xie
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Correspondence to Xiaoping Xie.

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This work was supported by National Natural Science Foundation of China (11771312).

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Han, Y., Chen, H., Wang, XP. et al. EXtended HDG Methods for Second Order Elliptic Interface Problems. J Sci Comput 84, 22 (2020). https://doi.org/10.1007/s10915-020-01272-3

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