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

Published online by Cambridge University Press:  09 July 2013

Blanca Ayuso de Dios ,
Ivan Georgiev ,
Johannes Kraus  and
Ludmil Zikatanov
Blanca Ayuso de Dios
Affiliation:
Centre de Recerca Matemàtica, Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain.. bayuso@crm.cat
Ivan Georgiev
Affiliation:
Institue of Mathematics and Informatics, Bulgarian Academy of Sciences and Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria.; ivan.georgiev@oeaw.ac.at
Johannes Kraus
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040 Linz, Austria.; johannes.kraus@oeaw.ac.at
Ludmil Zikatanov
Affiliation:
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.; ltz@math.psu.edu
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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.

Keywords

Linear elasticity equationslocking free discretizationspreconditioning
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1315 - 1333
Copyright
© EDP Sciences, SMAI, 2013

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References

Arnold, D.N., Brezzi, F., Cockburn, B. and Donatella Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001/02) 1749–1779. Google Scholar
Arnold, D.N., Brezzi, Franco, Falk, R. and Donatella Marini, L., Locking–free Reissner-Mindlin elements without reduced integration. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3660–3671. Google Scholar
Arnold, D.N., Brezzi, F. and Donatella Marini, L., A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput. 22-23 (2005) 25–45. Google Scholar
Ayuso de Dios, B. and Zikatanov, L., Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40 (2009) 4–36. Google Scholar
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.
Brezzi, F., Cockburn, B., Marini, L.D. and Süli, E., Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3293–3310. Google Scholar
Burman, E. and Stamm, B., Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2008) 508–533. Google Scholar
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).
Falk, R.S., Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529–550. Google Scholar
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.
Hansbo, P. and Larson, M.G., Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche’s method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895–1908. Google Scholar
Hansbo, P. and Larson, M.G., Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: M2AN 37 (2003) 63–72. Google Scholar
M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49 409–436 (1953), 1952.
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).
Y. Saad, Iterative methods for sparse linear systems. Society Industrial Appl. Math. Philadelphia, PA, 2nd (2003).
Wihler, T.P., Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45–75. Google Scholar
Wihler, T.P., Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comput. 75 (2006) 1087–1102. Google Scholar