A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1315-1333.

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.

DOI : 10.1051/m2an/2013070
Classification : 65F10, 65N20, 65N30
Mots-clés : linear elasticity equations, locking free discretizations, preconditioning 
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     title = {A subspace correction method for discontinuous {Galerkin} discretizations of linear elasticity equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1315--1333},
     publisher = {EDP-Sciences},
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     url = {http://www.numdam.org/articles/10.1051/m2an/2013070/}
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Ayuso de Dios, Blanca; Georgiev, Ivan; Kraus, Johannes; Zikatanov, Ludmil. A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 5, pp. 1315-1333. doi : 10.1051/m2an/2013070. http://www.numdam.org/articles/10.1051/m2an/2013070/

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[4] B. Ayuso De Dios and L. Zikatanov, Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40 (2009) 4-36. | MR | Zbl

[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] 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. | MR | Zbl

[7] E. Burman and B. Stamm, Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2008) 508-533. | MR | Zbl

[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). | MR | Zbl

[9] R.S. Falk, Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529-550. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[12] P. Hansbo and M.G. Larson, Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: M2AN 37 (2003) 63-72. | Numdam | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[15] Y. Saad, Iterative methods for sparse linear systems. Society Industrial Appl. Math. Philadelphia, PA, 2nd (2003). | MR | Zbl

[16] T.P. Wihler, Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45-75. | MR | Zbl

[17] T.P. Wihler, Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comput. 75 (2006) 1087-1102. | MR | Zbl

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