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 equations, locking free discretizations, preconditioning
@article{M2AN_2013__47_5_1315_0, author = {Ayuso de Dios, Blanca and Georgiev, Ivan and Kraus, Johannes and Zikatanov, Ludmil}, 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}, volume = {47}, number = {5}, year = {2013}, doi = {10.1051/m2an/2013070}, mrnumber = {3100765}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2013070/} }
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 PY - 2013 SP - 1315 EP - 1333 VL - 47 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013070/ DO - 10.1051/m2an/2013070 LA - en ID - M2AN_2013__47_5_1315_0 ER -
%0 Journal Article %A Ayuso de Dios, Blanca %A Georgiev, Ivan %A Kraus, Johannes %A Zikatanov, Ludmil %T A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 1315-1333 %V 47 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013070/ %R 10.1051/m2an/2013070 %G en %F M2AN_2013__47_5_1315_0
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 , Volume 47 (2013) no. 5, pp. 1315-1333. doi : 10.1051/m2an/2013070. http://www.numdam.org/articles/10.1051/m2an/2013070/
[1] Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39 (2001/02) 1749-1779. | MR | Zbl
, , and ,[2] 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] A family of discontinuous Galerkin finite elements for the Reissner-Mindlin plate. J. Sci. Comput. 22-23 (2005) 25-45. | MR | Zbl
, and ,[4] Uniformly convergent iterative methods for discontinuous Galerkin discretizations. J. Sci. Comput. 40 (2009) 4-36. | MR | Zbl
and ,[5] 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.
, and ,[6] Stabilization mechanisms in discontinuous Galerkin finite element methods. Comput. Methods Appl. Mech. Engrg. 195 (2006) 3293-3310. | MR | Zbl
, , and ,[7] Low order discontinuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2008) 508-533. | MR | Zbl
and ,[8] Inequalities in mechanics and physics. Springer-Verlag, Berlin, Translated from the French by C.W. John, Grundlehren der Mathematischen Wissenschaften 219 (1976). | MR | Zbl
and ,[9] Nonconforming finite element methods for the equations of linear elasticity. Math. Comput. 57 (1991) 529-550. | MR | Zbl
,[10] 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
, and ,[11] Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1895-1908. | MR | Zbl
and ,[12] Discontinuous Galerkin and the Crouzeix-Raviart element: application to elasticity. ESAIM: M2AN 37 (2003) 63-72. | Numdam | MR | Zbl
and ,[13] Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards 49 409-436 (1953), 1952. | MR | Zbl
and ,[14] Robust algebraic multilevel methods and algorithms. Walter de Gruyter GmbH and Co. KG, Berlin. Radon Ser. Comput. Appl. Math. 5 (2009). | MR | Zbl
and ,[15] Iterative methods for sparse linear systems. Society Industrial Appl. Math. Philadelphia, PA, 2nd (2003). | MR | Zbl
,[16] Locking-free DGFEM for elasticity problems in polygons. IMA J. Numer. Anal. 24 (2004) 45-75. | MR | Zbl
,[17] Locking-free adaptive discontinuous Galerkin FEM for linear elasticity problems. Math. Comput. 75 (2006) 1087-1102. | MR | Zbl
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