A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry
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 \(k \ge d\). Finally, we also provide numerical results of MDG showing that the orders of convergence are actually sharp.
- Authors:
-
[2];
- Xi'an Jiaotong Univ. (China). State Key Lab. of Multiphase Flow in Power Engineering
- Peking Univ., Beijing (China)
- Pennsylvania State Univ., University Park, PA (United States)
- Publication Date:
- Research Org.:
- Pennsylvania State Univ., University Park, PA (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1802634
- Grant/Contract Number:
- SC0014400
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Scientific Computing
- Additional Journal Information:
- Journal Volume: 83; Journal Issue: 1; Journal ID: ISSN 0885-7474
- Publisher:
- Springer
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Mixed DG method; Linear elasticity; Well-posedness; A priori error analysis
Citation Formats
Wang, Fei, Wu, Shuonan, and Xu, Jinchao. A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry. United States: N. p., 2020.
Web. doi:10.1007/s10915-020-01191-3.
Wang, Fei, Wu, Shuonan, & Xu, Jinchao. A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry. United States. https://doi.org/10.1007/s10915-020-01191-3
Wang, Fei, Wu, Shuonan, and Xu, Jinchao. Tue .
"A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry". United States. https://doi.org/10.1007/s10915-020-01191-3. https://www.osti.gov/servlets/purl/1802634.
@article{osti_1802634,
title = {A Mixed Discontinuous Galerkin Method for Linear Elasticity with Strongly Imposed Symmetry},
author = {Wang, Fei and Wu, Shuonan and Xu, Jinchao},
abstractNote = {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 \(k \ge d\). Finally, we also provide numerical results of MDG showing that the orders of convergence are actually sharp.},
doi = {10.1007/s10915-020-01191-3},
journal = {Journal of Scientific Computing},
number = 1,
volume = 83,
place = {United States},
year = {Tue Mar 17 00:00:00 EDT 2020},
month = {Tue Mar 17 00:00:00 EDT 2020}
}
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