Abstract
In this work, we construct energy-minimizing coarse spaces for the finite element discretization of mixed boundary value problems for displacements in compressible linear elasticity. Motivated from the multiscale analysis of highly heterogeneous composite materials, basis functions on a triangular coarse mesh are constructed, obeying a minimal energy property subject to global pointwise constraints. These constraints allow that the coarse space exactly contains the rigid body translations, while rigid body rotations are preserved approximately. The application is twofold. Resolving the heterogeneities on the finest scale, we utilize the energy-minimizing coarse space for the construction of robust two-level overlapping domain decomposition preconditioners. Thereby, we do not assume that coefficient jumps are resolved by the coarse grid, nor do we impose assumptions on the alignment of material jumps and the coarse triangulation. Weonly assume that the size of the inclusions is small compared to the coarse mesh diameter. Ournumerical tests show uniform convergence rates independent of the contrast in the Young’s modulus within the heterogeneous material. Furthermore, we numerically observe the properties of the energy-minimizing coarse space in an upscaling framework. Therefore, we present numerical results showing the approximation errors of the energy-minimizing coarse space w.r.t. the fine-scale solution.
Mathematics Subject Classification (2010): 35R05, 65F10, 65F10, 65N22, 65N55, 74B05, 74S05
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Baker, A.H., Kolev, T., Yang, U.M.: Improving algebraic multigrid interpolation operators for linear elasticity problems. Numer. Lin. Algebra Appl. 17, 495–517 (2010)
Braess, D.: Finite Elements: Theory, Fast Solvers and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)
Brezina, M., Cleary, A.J., Falgout, R.D., Henson, V.E., Jones, J.E., Manteuffel, T.A., Mccormick, S.F., Ruge, J.W.: Algebraic multigrid based on element interpolation (AMGe). SIAM J. Sci. Comput. 22, 1570–1592 (2000)
Buck, M., Iliev, O., Andrä, H.: Multiscale finite element coarse spaces for the application to linear elasticity. Cent. Eur. J. Math. 11(4), 680–701 (2013)
Clees, T.: AMG strategies for PDE systems with applications in industrial semiconductor simulation. Thesis, Faculty of Mathematics, University of Cologne (2005)
Dohrmann, C., Klawonn, A., Widlund, O.: A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners. Domain Decomposition Methods in Science and Engineering XVII, Springer, 247–254 (2008)
Efendiev, Y., Galvis, J., Lazarov, R., Willems, J.: Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal. 46, 1175–1199 (2012)
Graham, I.G., Lechner, P.O., Scheichl, R.: Domain decomposition for multiscale PDEs. Numer. Math. 106, 589–626 (2007)
Janka, A.: Algebraic domain decomposition solver for linear elasticity. Appl. Math. 44, 435–458 (1999)
Jones, J., Vassilevski, P.S.: AMGe based on element agglomeration. SIAM J. Sci. Comput. 23, 109–133 (2001)
Karer, E.: Subspace correction methods for linear elasticity. Thesis, University of Linz (2011)
Karer, E., Kraus, J.K.: Algebraic multigrid for finite element elasticity equations: determination of nodal dependence via edge matrices and two-level convergence. Int. J. Numer. Meth. Eng. 83, 642–670 (2010)
Kolev, T.V., Vassilevski, P.S.: AMG by element agglomeration and constrained energy minimization interpolation. Numer. Lin. Algebra Appl. 13, 771–788 (2006)
Kraus, J.K.: Algebraic multigrid based on computational molecules, II: linear elasticity problems. SIAM J. Sci. Comput. 30, 505–524 (2008)
Kraus, J.K., Schicho, J.: Algebraic multigrid based on computational molecules I: scalar elliptic problems. Computing 77, 57–75 (2006)
Mandel, J., Brezina, M., Vaněk, P.: Energy optimization of algebraic multigrid bases. Computing 62, 205–228 (1999)
Olson, L.N., Schroder, J.B., Tuminaro, R.S.: A general interpolation strategy for algebraic multigrid using energy minimization. SIAM J. Sci. Comput. 33, 966–991 (2011)
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Sarkis, M.: Partition of unity coarse spaces: enhanced versions, discontinuous coefficients and applications to elasticity. In: Domain Decomposition Methods in Science and Engineering XIV., Natl. Auton. Univ. Mex., Mexico, 149–158 (2003)
Scheichl, R., Vassilevski, P.S., Zikatanov, L.T.: Weak approximation properties of elliptic projections with functional constraints. Multiscale Model. Simul. 9, 1677–1699 (2011)
Schulz, V., Andrä, H., Schmidt, K.: Robuste Netzgenerierung zur Mikro-FE-Analyse mikrostrukturierter Materialien. In: NAFEMS Magazin, vol. 2, pp. 28–30 (2007)
Smith, B.F.: Domain decomposition algorithms for the partial differential equations of linear elasticity. Thesis, Courant Institute of Mathematical Sciences, New York University (1990)
Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C., Scheichl, R.: Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps. Technical Report 2011–07, University of Linz, Institute of Computational Mathematics (2011)
Toselli, A., Widlund, O.: Domain Decomposition Methods, Algorithms and Theory. Springer, Berlin (2005)
Van lent, J., Scheichl, R., Graham, I.G.: Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. Numer. Lin. Algebra Appl. 16, 775–799 (2009)
Vaněk, P., Brezina, M., Tezaur, R.: Two-grid method for linear elasticity on unstructured meshes. SIAM J. Sci. Comput. 21, 900–923 (1999)
Vassilevski, P.S.: Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations. Springer, New York (2008)
Vassilevski, P.S.: General constrained energy minimizing interpolation mappings for AMG. SIAM J. Sci. Comput. 32, 1–13 (2010)
Wan, W., Chan, T.F., Smith, B.: An energy-minimizing interpolation for robust multigrid methods. SIAM J. Sci. Comput. 21, 1632–1649 (2000)
Willems, J.: Robust multilevel methods for general symmetric positive definite operators. Technical Report 2012–06, RICAM Institute for Computational and Applied Mathematics (2012)
Xu, J., Zikatanov, L.T.: On an energy minimizing basis in algebraic multigrid methods. Comput. Vis. Sci. 7, 121–127 (2004)
Acknowledgements
The authors would like to thank Dr. Panayot Vassilevski and Prof. Ludmil Zikatanov for many fruitful discussions and their valuable comments on the subject of this paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Buck, M., Iliev, O., Andrä, H. (2013). Multiscale Coarsening for Linear Elasticity by Energy Minimization. In: Iliev, O., Margenov, S., Minev, P., Vassilevski, P., Zikatanov, L. (eds) Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications. Springer Proceedings in Mathematics & Statistics, vol 45. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7172-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7172-1_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7171-4
Online ISBN: 978-1-4614-7172-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)