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Discontinuous Galerkin Method with Staggered Hybridization for a Class of Nonlinear Stokes Equations

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Abstract

In this paper, we present a discontinuous Galerkin method with staggered hybridization to discretize a class of nonlinear Stokes equations in two dimensions. The utilization of staggered hybridization is new and this approach combines the features of traditional hybridization method and staggered discontinuous Galerkin method. The main idea of our method is to use hybrid variables to impose the staggered continuity conditions instead of enforcing them in the approximation space. Therefore, our method enjoys some distinctive advantages, including mass conservation, optimal convergence and preservation of symmetry of the stress tensor. We will also show that, one can obtain superconvergent and strongly divergence-free velocity by applying a local postprocessing technique on the approximate solution. We will analyze the stability and derive a priori error estimates of the proposed scheme. The resulting nonlinear system is solved by using the Newton’s method, and some numerical results will be demonstrated to confirm the theoretical rates of convergence and superconvergence.

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Acknowledgements

The research of Eric Chung and Xiao-Ping Wang is partially supported by a NSFC/RGC Joint Research Scheme (Project: N_HKUST620/15). The research of Eric Chung is partially supported by Hong Kong RGC General Research Fund (Project number: 14317516).

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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 T. Chung & Ming Fai Lam

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

    Xiao-Ping Wang

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  1. Jie Du
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  2. Eric T. Chung
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  3. Ming Fai Lam
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  4. Xiao-Ping Wang
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Correspondence to Jie Du.

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Du, J., Chung, E.T., Lam, M.F. et al. Discontinuous Galerkin Method with Staggered Hybridization for a Class of Nonlinear Stokes Equations. J Sci Comput 76, 1547–1577 (2018). https://doi.org/10.1007/s10915-018-0676-z

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  • DOI: https://doi.org/10.1007/s10915-018-0676-z

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