Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (2): 477-499.doi: 10.1007/s42967-021-00128-3

• ORIGINAL PAPERS • 上一篇    下一篇

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods

Gang Chen1, Bernardo Cockburn2, John R. Singler3, Yangwen Zhang4   

  1. 1 College of Mathematics, Sichuan University, Chengdu 610065, Sichuan, China;
    2 School of Mathematics, University of Minnesota, Minneapolis, MN, USA;
    3 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA;
    4 Department of Mathematical Science, University of Delaware, Newark, DE, USA
  • 收稿日期:2020-08-31 修回日期:2021-02-01 出版日期:2022-06-20 发布日期:2022-04-29
  • 通讯作者: Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang E-mail:cglwdm@scu.edu.cn;cockburn@math.umn.edu;singlerj@mst.edu;ywzhangf@udel.edu
  • 基金资助:
    G. Chen was supported by the National Natural Science Foundation of China (NSFC) Grant 11801063, and the Fundamental Research Funds for the Central Universities Grant YJ202030. B. Cockburn was partially supported by the National Science Foundation Grant DMS-1912646. J. Singler and Y. Zhang were supported in part by the National Science Foundation Grant DMS-1217122.

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods

Gang Chen1, Bernardo Cockburn2, John R. Singler3, Yangwen Zhang4   

  1. 1 College of Mathematics, Sichuan University, Chengdu 610065, Sichuan, China;
    2 School of Mathematics, University of Minnesota, Minneapolis, MN, USA;
    3 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA;
    4 Department of Mathematical Science, University of Delaware, Newark, DE, USA
  • Received:2020-08-31 Revised:2021-02-01 Online:2022-06-20 Published:2022-04-29
  • Contact: Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang E-mail:cglwdm@scu.edu.cn;cockburn@math.umn.edu;singlerj@mst.edu;ywzhangf@udel.edu
  • Supported by:
    G. Chen was supported by the National Natural Science Foundation of China (NSFC) Grant 11801063, and the Fundamental Research Funds for the Central Universities Grant YJ202030. B. Cockburn was partially supported by the National Science Foundation Grant DMS-1912646. J. Singler and Y. Zhang were supported in part by the National Science Foundation Grant DMS-1217122.

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摘要： In Chen et al. (J. Sci. Comput. 81(3):2188-2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defned on simplicial meshes, for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k ≥ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3):635-650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defned on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k ≥ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

关键词: Hybrid high-order methods, Hybridizable discontinuous Galerkin methods, Interpolatory method, Superconvergence

Abstract: In Chen et al. (J. Sci. Comput. 81(3):2188-2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defned on simplicial meshes, for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k ≥ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3):635-650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defned on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k ≥ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory.

Key words: Hybrid high-order methods, Hybridizable discontinuous Galerkin methods, Interpolatory method, Superconvergence

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