A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗

Yongxing Shen; Adrian J. Lew

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 5, page 1003-1028
  • ISSN: 0764-583X

Abstract

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We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent of the value of the Poisson’s ratio. These estimates demonstrate that these methods are locking-free. To this end, we prove the corresponding inf-sup condition, which for the equal-order case, requires a construction to establish the surjectivity of the space of discrete divergences on the pressure space. In the particular case of near incompressibility and equal-order approximation of the displacement and pressure fields, the mixed method is equivalent to a displacement method proposed earlier by Lew et al. [Appel. Math. Res. express 3 (2004) 73–106]. The absence of locking of this displacement method then follows directly from that of the mixed method, including the uniform error estimate for the stress with respect to the Poisson’s ratio. We showcase the performance of these methods through numerical examples, which show that locking may appear if Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

How to cite

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Shen, Yongxing, and Lew, Adrian J.. "A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1003-1028. <http://eudml.org/doc/277848>.

@article{Shen2012,
abstract = {We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent of the value of the Poisson’s ratio. These estimates demonstrate that these methods are locking-free. To this end, we prove the corresponding inf-sup condition, which for the equal-order case, requires a construction to establish the surjectivity of the space of discrete divergences on the pressure space. In the particular case of near incompressibility and equal-order approximation of the displacement and pressure fields, the mixed method is equivalent to a displacement method proposed earlier by Lew et al. [Appel. Math. Res. express 3 (2004) 73–106]. The absence of locking of this displacement method then follows directly from that of the mixed method, including the uniform error estimate for the stress with respect to the Poisson’s ratio. We showcase the performance of these methods through numerical examples, which show that locking may appear if Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.},
author = {Shen, Yongxing, Lew, Adrian J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Discontinuous Galerkin; locking; mixed method; inf-sup condition; discontinuous Galerkin},
language = {eng},
month = {2},
number = {5},
pages = {1003-1028},
publisher = {EDP Sciences},
title = {A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗},
url = {http://eudml.org/doc/277848},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Shen, Yongxing
AU - Lew, Adrian J.
TI - A family of discontinuous Galerkin mixed methods for nearly and perfectly incompressible elasticity∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/2//
PB - EDP Sciences
VL - 46
IS - 5
SP - 1003
EP - 1028
AB - We introduce a family of mixed discontinuous Galerkin (DG) finite element methods for nearly and perfectly incompressible linear elasticity. These mixed methods allow the choice of polynomials of any order k ≥ 1 for the approximation of the displacement field, and of order k or k − 1 for the pressure space, and are stable for any positive value of the stabilization parameter. We prove the optimal convergence of the displacement and stress fields in both cases, with error estimates that are independent of the value of the Poisson’s ratio. These estimates demonstrate that these methods are locking-free. To this end, we prove the corresponding inf-sup condition, which for the equal-order case, requires a construction to establish the surjectivity of the space of discrete divergences on the pressure space. In the particular case of near incompressibility and equal-order approximation of the displacement and pressure fields, the mixed method is equivalent to a displacement method proposed earlier by Lew et al. [Appel. Math. Res. express 3 (2004) 73–106]. The absence of locking of this displacement method then follows directly from that of the mixed method, including the uniform error estimate for the stress with respect to the Poisson’s ratio. We showcase the performance of these methods through numerical examples, which show that locking may appear if Dirichlet boundary conditions are imposed strongly rather than weakly, as we do here.
LA - eng
KW - Discontinuous Galerkin; locking; mixed method; inf-sup condition; discontinuous Galerkin
UR - http://eudml.org/doc/277848
ER -

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