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A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry

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Abstract

In this paper, we study a mixed discontinuous Galerkin (MDG) method to solve linear elasticity problem with arbitrary order discontinuous finite element spaces in d-dimension (\(d=2,3\)). This method uses polynomials of degree \(k+1\) for the stress and of degree k for the displacement (\(k\ge 0\)). The mixed DG scheme is proved to be well-posed under proper norms. Specifically, we prove that, for any \(k \ge 0\), the \(H(\mathrm{div})\)-like error estimate for the stress and \(L^2\) error estimate for the displacement are optimal. We further establish the optimal \(L^2\) error estimate for the stress provided that the \({\mathcal {P}}_{k+2}-{\mathcal {P}}_{k+1}^{-1}\) Stokes pair is stable and \(k \ge d\). We also provide numerical results of MDG showing that the orders of convergence are actually sharp.

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Correspondence to Shuonan Wu.

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The work of Fei Wang is partially supported by the National Natural Science Foundation of China (Grant No. 11771350). The work of Shuonan Wu is partially supported by the National Natural Science Foundation of China (Grant No. 11901016) and the startup Grant from Peking University. The work of the Jinchao Xu is partially supported by US Department of Energy Grant DE-SC0014400 and National Science Foundation Grant DMS-1819157.

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Wang, F., Wu, S. & Xu, J. A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry. J Sci Comput 83, 2 (2020). https://doi.org/10.1007/s10915-020-01191-3

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  • DOI: https://doi.org/10.1007/s10915-020-01191-3

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