New discontinuous Galerkin algorithms and analysis for linear elasticity with symmetric stress tensor

Qingguo Hong, Jun Hu, Limin Ma, Jinchao Xu

Research output: Contribution to journalArticlepeer-review


This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor σh and displacement uh inside each element, and the modifications of these two variables σˇ h and uˇ h on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as H1-based and the other known as H(div) -based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.
Original languageEnglish (US)
Pages (from-to)645-678
Number of pages34
JournalNumerische Mathematik
Issue number3
StatePublished - Nov 1 2021
Externally publishedYes

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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