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A Hybridizable Discontinuous Galerkin Method for Kirchhoff Plates

Authors
  • Huang, Jianguo1
  • Huang, Xuehai2, 3
  • 1 Shanghai Jiao Tong University, Key Laboratory of Scientific and Engineering Computing (Ministry of Education), School of Mathematical Sciences, Shanghai, 200240, China , Shanghai (China)
  • 2 Shanghai University of Finance and Economics, School of Mathematics, Shanghai, 200433, China , Shanghai (China)
  • 3 Wenzhou University, Department of Mathematics, Wenzhou, 325035, China , Wenzhou (China)
Type
Published Article
Journal
Journal of Scientific Computing
Publisher
Springer US
Publication Date
Jul 07, 2018
Volume
78
Issue
1
Pages
290–320
Identifiers
DOI: 10.1007/s10915-018-0780-0
Source
Springer Nature
Keywords
License
Yellow

Abstract

With the introduction of numerical traces respectively related to the normal bending moment, the twisting moment and the effective transverse shear force, and based on the Hermann–Miyoshi formulation, this paper proposes a hybridizable discontinuous Galerkin (HDG) method for Kirchhoff plate bending problems. The piecewise polynomials of degrees k-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k-1$$\end{document} and k are used to approximate the moment and the deflection, respectively. The optimal and superconvergent error estimates are derived under minimal regularity assumptions on the exact solution. The key ingredients in the analysis include the derivation of a discrete inf-sup condition and some local lower bound estimates of a posteriori error analysis. The significant feature of the HDG method is superconvergence as well as the low number of globally coupled degrees of freedom associated with Lagrange multipliers. Furthermore, a new discrete deflection is constructed by postprocessing the solution of the HDG method, which superconverges to the deflection with order k+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+1$$\end{document} in broken H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document} norm. Finally, some numerical results are shown to demonstrate the theoretical results.

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