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

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Abstract

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 \(\varvec{\sigma }_h\) and displacement \(u_h\) inside each element, and the modifications of these two variables \(\check{\varvec{\sigma }}_h\) and \({\check{u}}_h\) 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 \(H^1\)-based and the other known as \(H(\mathrm{div})\)-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.

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Qingguo Hong, Limin Ma & Jinchao Xu

  2. School of Mathematical Science, Peking University, Beijing, 100871, People’s Republic of China

    Jun Hu

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  1. Qingguo Hong
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  2. Jun Hu
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Correspondence to Limin Ma.

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The work of Hong, Ma and Xu was partially supported by Center for Computational Mathematics and Applications, The Pennsylvania State University. The work of Hu was supported by NSFC projects 11625101.

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Hong, Q., Hu, J., Ma, L. et al. New discontinuous Galerkin algorithms and analysis for linear elasticity with symmetric stress tensor. Numer. Math. 149, 645–678 (2021). https://doi.org/10.1007/s00211-021-01234-3

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  • DOI: https://doi.org/10.1007/s00211-021-01234-3

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