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 -

References

top
  1. D.N. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations. Calcolo21 (1984) 337–344.  Zbl0593.76039
  2. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2002) 1749–1779.  Zbl1008.65080
  3. F. Bassi and S. Rebay, A High-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys.131 (1997) 267–279.  Zbl0871.76040
  4. R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity. Comput. Methods Appl. Mech. Engrg.198 (2009) 3352–3360.  Zbl1230.74169
  5. M. Bercovier and O.A. Pironneau, Error estimates for finite element method solution of the Stokes problem in the primitive variables. Numer. Math.33 (1977) 211–224.  Zbl0423.65058
  6. S.C. Brenner, Korn’s inequalities for piecewise H1 vector fields. Math. Comp.73 (2003) 1067–1087.  Zbl1055.65118
  7. S.C. Brenner, Poincaré-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal.41 (2003) 306–324.  Zbl1045.65100
  8. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3th edition, Springer (2008).  Zbl1135.65042
  9. S.C. Brenner and L.-Y. Sung, Linear finite element methods for planar linear elasticity. Math. Comp.59 (1992) 321–338.  Zbl0766.73060
  10. F. Brezzi, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numér.8 (1974) 129–151.  Zbl0338.90047
  11. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods. Springer Series in Computational Mathematics, Springer-Verlag, New York (1991).  Zbl0788.73002
  12. F. Brezzi, J. DouglasJr., and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217–235.  Zbl0599.65072
  13. F. Brezzi, J. DouglasJr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables. Numer. Math.51 (1987) 237–250.  
  14. F. Brezzi, G. Manzini, D. Marini, P. Pietra and A. Russo, Discontinuous galerkin approximations for elliptic problems. Numer. Methods Partial Differential Equations (2000) 365–378.  Zbl0957.65099
  15. F. Brezzi, T.J.R. Hughes, L.D. Marini and A. Masud, Mixed discontinuous Galerkin methods for Darcy flow. J. Sci. Comput.22, 23 (2005) 119–145.  Zbl1103.76031
  16. J. Carrero, B. Cockburn and D. Schötzau, Hybridized globally divergence-free LDG methods. Part I : the Stokes problem. Math. Comp.75 (2005) 533–563.  Zbl1087.76061
  17. P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).  Zbl0383.65058
  18. B. Cockburn, G. Kanschat, D. Schötzau and C. Schwab, Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal.40 (2002) 319–343.  Zbl1032.65127
  19. B. Cockburn, D. Schötzau and J. Wang, Discontinuous Galerkin methods for incompressible elastic materials. Comput. Methods Appl. Mech. Engrg.195 (2006) 3184–3204.  Zbl1128.74041
  20. M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO Sér. Rouge7 (1973) 33–75.  Zbl0302.65087
  21. M. Fortin, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér.11 (1977) 341–354.  Zbl0373.65055
  22. V. Girault, B. Rivière and M.F. Wheeler, A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems. Math. Comp.74 (2005) 53–84.  Zbl1057.35029
  23. P. Hansbo and M.G. Larson, Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Engrg.191 (2002) 1895–1908.  Zbl1098.74693
  24. P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity. ESAIM : M2AN37 (2003) 63–72.  Zbl1137.65431
  25. P. Hansbo and M.G. Larson, Piecewise divergence-free discontinuous Galerkin methods for Stokes flow. Comm. Num. Methods Engrg.24 (2008) 355–366.  Zbl1138.76046
  26. F. Hecht, Construction d’une base de fonctions P1 non conforme à divergence nulle dans R3. RAIRO Anal. Numér.15 (1981) 119–150.  Zbl0471.76028
  27. P. Hood and C. Taylor, Numerical solution of the Navier-Stokes equations using the finite element technique. Comput. Fluids1 (1973) 1–28.  Zbl0328.76020
  28. P. Hood and C. Taylor, Navier-Stokes equations using mixed interpolation. Finite Element Methods in Flow Problems, edited by J.T. Oden. UAH Press, Huntsville, Alabama (1974).  
  29. R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Engrg.124 (1995) 195–212.  Zbl1067.74578
  30. A. Lew, P. Neff, D. Sulsky and M. Ortiz, Optimal BV estimates for a discontinuous Galerkin method for linear elasticity. Appl. Math. Res. express3 (2004) 73–106.  Zbl1115.74021
  31. N.C. Nguyen, J. Peraire and B. Cockburn, A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Engrg.199 (2010) 582–597.  Zbl1227.76036
  32. B. Rivière and V. Girault, Discontinuous finite element methods for incompressible flows on subdomains with non-matching interfaces. Comput. Methods Appl. Mech. Engrg.195 (2006) 3274–3292.  Zbl1121.76038
  33. D. Schötzau, C. Schwab and A. Toselli, Mixed hp-DGFEM for incompressible flows. SIAM J. Numer. Anal.40 (2003) 2171–2194.  Zbl1055.76032
  34. L.R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Modélisation Mathématique et Analyse Numérique19 (1985) 111–143.  Zbl0608.65013
  35. S.-C. Soon, B. Cockburn and H.K. Stolarski, A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Engrg.80 (2009), 1058–1092.  Zbl1176.74196
  36. A. Ten Eyck, and A. Lew, Discontinuous Galerkin methods for non-linear elasticity. Int. J. Numer. Methods Engrg.67 (2006) 1204–1243.  Zbl1113.74068
  37. A. Ten Eyck, F. Celiker and A. Lew, Adaptive stabilization of discontinuous Galerkin methods for nonlinear elasticity : Analytical estimates. Comput. Methods Appl. Mech. Engrg.197 (2008) 2989–3000.  Zbl1194.74390
  38. F. Thomasset, Implementation of finite element methods for Navier-Stokes equations. Springer-Verlag, New York (1981).  Zbl0475.76036
  39. J.P. Whiteley, Discontinuous Galerkin finite element methods for incompressible non-linear elasticity, Comput. Methods Appl. Mech. Engrg.198 (2009) 3464–3478.  Zbl1230.74200
  40. T.P. Wihler, Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal.24 (2004) 45–75.  Zbl1057.74046

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