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

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

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

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topShen, 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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