A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

Blanca Ayuso de Dios; Ivan Georgiev; Johannes Kraus; Ludmil Zikatanov

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 5, page 1315-1333
  • ISSN: 0764-583X

Abstract

top
We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

How to cite

top

Ayuso de Dios, Blanca, et al. "A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.5 (2013): 1315-1333. <http://eudml.org/doc/273141>.

@article{AyusodeDios2013,
abstract = {We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.},
author = {Ayuso de Dios, Blanca, Georgiev, Ivan, Kraus, Johannes, Zikatanov, Ludmil},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {linear elasticity equations; locking free discretizations; preconditioning},
language = {eng},
number = {5},
pages = {1315-1333},
publisher = {EDP-Sciences},
title = {A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations},
url = {http://eudml.org/doc/273141},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Ayuso de Dios, Blanca
AU - Georgiev, Ivan
AU - Kraus, Johannes
AU - Zikatanov, Ludmil
TI - A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 5
SP - 1315
EP - 1333
AB - We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.
LA - eng
KW - linear elasticity equations; locking free discretizations; preconditioning
UR - http://eudml.org/doc/273141
ER -

References

top
  1. [1] D.N. Arnold, F. Brezzi, B. Cockburn and L. Donatella Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001/02) 1749–1779. Zbl1008.65080MR1885715
  2. [2] D.N. Arnold, Franco Brezzi, R.Falk and L. Donatella Marini, Locking–free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3660–3671. Zbl1173.74396MR2339992
  3. [3] D.N. Arnold, F. Brezzi and L. Donatella Marini, A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput. 22-23 (2005) 25–45. Zbl1065.74068MR2142189
  4. [4] B. Ayuso de Dios and L. Zikatanov, Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput.40 (2009) 4–36. Zbl1203.65242MR2511726
  5. [5] R. Blaheta, S. Margenov and M. Neytcheva, Aggregation-based multilevel preconditioning of non-conforming fem elasticity problems. Applied Parallel Computing. State of the Art in Scientific Computing, edited by J. Dongarra, K. Madsen and J. Wasniewski. In Lect. Notes Comput. Sci., vol. 3732. Springer Berlin/Heidelberg (2006) 847–856. 
  6. [6] F. Brezzi, B. Cockburn, L.D. Marini and E. Süli, Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg.195 (2006) 3293–3310. Zbl1125.65102MR2220920
  7. [7] E. Burman and B. Stamm, Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal.47 (2008) 508–533. Zbl1190.65170MR2475950
  8. [8] G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics. Springer-Verlag, Berlin, Translated from the French by C.W. John, Grundlehren der Mathematischen Wissenschaften 219 (1976). Zbl0331.35002MR521262
  9. [9] R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput.57 (1991) 529–550. Zbl0747.73044MR1094947
  10. [10] I. Georgiev, J.K. Kraus and S. Margenov, Multilevel preconditioning of Crouzeix-Raviart 3D pure displacement elasticity problems. Large Scale Scientific Computing, edited by I. Lirkov, S. Margenov and J. Wasniewski. In Lect. Notes Comput. Science, vol. 5910. Springer, Berlin, Heidelberg (2010) 103–110. Zbl1280.74036
  11. [11] 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. Zbl1098.74693MR1886000
  12. [12] P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: M2AN 37 (2003) 63–72. Zbl1137.65431MR1972650
  13. [13] M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49 409–436 (1953), 1952. Zbl0048.09901MR60307
  14. [14] J. Kraus and S. Margenov, Robust algebraic multilevel methods and algorithms. Walter de Gruyter GmbH and Co. KG, Berlin. Radon Ser. Comput. Appl. Math. 5 (2009). Zbl1184.65113MR2574100
  15. [15] Y. Saad, Iterative methods for sparse linear systems. Society Industrial Appl. Math. Philadelphia, PA, 2nd (2003). Zbl1031.65046MR1990645
  16. [16] T.P. Wihler, Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal.24 (2004) 45–75. Zbl1057.74046MR2027288
  17. [17] T.P. Wihler, Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comput.75 (2006) 1087–1102. Zbl1088.74047MR2219020

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.