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An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation

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Abstract

Staggered grid techniques have been applied successfully to many problems. A distinctive advantage is that physical laws arising from the corresponding partial differential equations are automatically preserved. Recently, a staggered discontinuous Galerkin (SDG) method was developed for the convection–diffusion equation. In this paper, we are interested in solving the steady state convection–diffusion equation with a small diffusion coefficient \(\epsilon \). It is known that the exact solution may have large gradient in some regions and thus a very fine mesh is needed. For convection dominated problems, that is, when \(\epsilon \) is small, exact solutions may contain sharp layers. In these cases, adaptive mesh refinement is crucial in order to reduce the computational cost. In this paper, a new SDG method is proposed and the proof of its stability is provided. In order to construct an adaptive mesh refinement strategy for this new SDG method, we derive an a-posteriori error estimator and prove its efficiency and reliability under a boundedness assumption on \(h/\epsilon \), where h is the mesh size. Moreover, we will present some numerical results with singularities and sharp layers to show the good performance of the proposed error estimator as well as the adaptive mesh refinement strategy.

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Acknowledgements

The work of Eric Chung is partially supported by Hong Kong RGC General Research Fund (Projects: 14317516, 14301314) and CUHK Direct Grant for Research 2016-17.

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Authors and Affiliations

  1. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Jie Du

  2. Department of Mathematics, The Chinese University of Hong Kong, Sha Tin, Hong Kong SAR

    Eric Chung

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  1. Jie Du
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Correspondence to Eric Chung.

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Du, J., Chung, E. An Adaptive Staggered Discontinuous Galerkin Method for the Steady State Convection–Diffusion Equation. J Sci Comput 77, 1490–1518 (2018). https://doi.org/10.1007/s10915-018-0695-9

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  • DOI: https://doi.org/10.1007/s10915-018-0695-9

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