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2013 | 11 | 4 | 680-701
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Multiscale finite element coarse spaces for the application to linear elasticity

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We extend the multiscale finite element method (MsFEM) as formulated by Hou and Wu in [Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169–189] to the PDE system of linear elasticity. The application, motivated by the multiscale analysis of highly heterogeneous composite materials, is twofold. Resolving the heterogeneities on the finest scale, we utilize the linear MsFEM basis for the construction of robust coarse spaces in the context of two-level overlapping domain decomposition preconditioners. We motivate and explain the construction and show that the constructed multiscale coarse space contains all the rigid body modes. Under the assumption that the material jumps are isolated, that is they occur only in the interior of the coarse grid elements, our numerical experiments show uniform convergence rates independent of the contrast in Young’s modulus within the heterogeneous material. Elsewise, if no restrictions on the position of the high coefficient inclusions are imposed, robustness cannot be guaranteed any more. These results justify expectations to obtain coefficient-explicit condition number bounds for the PDE system of linear elasticity similar to existing ones for scalar elliptic PDEs as given in the work of Graham, Lechner and Scheichl [Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589–626]. Furthermore, we numerically observe the properties of the MsFEM coarse space for linear elasticity in an upscaling framework. Therefore, we present experimental results showing the approximation errors of the multiscale coarse space w.r.t. the fine-scale solution.
  • [1] Baker A.H., Kolev Tz.V., Yang U.M., Improving algebraic multigrid interpolation operators for linear elasticity problems, Numer. Linear Algebra Appl., 2010, 17(2–3), 495–517
  • [2] Braess D., Finite Elements, 3rd ed., Cambridge University Press, Cambridge, 2007
  • [3] Brezzi F., Fortin M., Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math., 15, Springer, New York, 1991
  • [4] Chu C.-C., Graham I.G., Hou T.-Y., A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 2010, 79(272), 1915–1955
  • [5] Clees T., AMG Strategies for PDE Systems with Applications in Industrial Semiconductor Simulation, PhD thesis, Universität zu Köln, 2005, available at
  • [6] Clément Ph., Approximation by finite element functions using local regularization, RAIRO Anal. Numér., 1975, R-2, 77–84
  • [7] Durlofsky L.J., Efendiev Y., Ginting V., An adaptive local-global multiscale finite volume element method for twophase flow simulations, Adv. in Water Res., 2007, 30(3), 576–588
  • [8] Efendiev Y., Hou T.Y., Multiscale Finite Element Methods, Surv. Tutor. Appl. Math. Sci., 4, Springer, New York, 2009
  • [9] Efendiev Y., Galvis J., Lazarov R., Willems J., Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms, ESAIM Math. Model. Numer. Anal., 2012, 46(5), 1175–1199
  • [10] Falk R.S., Nonconforming finite element methods for the equations of linear elasticity, Math. Comp., 1991, 57(196), 529–550
  • [11] Galvis J., Efendiev Y., Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul., 2010, 8(4), 1461–1483
  • [12] Galvis J., Efendiev Y., Domain decomposition preconditioners for multiscale flows in high contrast media: reduced dimension coarse spaces, Multiscale Model. Simul., 2010, 8(5), 1621–1644
  • [13] Graham I.G., Lechner P.O., Scheichl R., Domain decomposition for multiscale PDEs, Numer. Math., 2007, 106(4), 589–626
  • [14] Graham I.G., Scheichl R., Robust domain decomposition algorithms for multiscale PDEs, Numer. Methods Partial Differential Equations, 2007, 23(4), 859–878
  • [15] Hou T.Y., Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 1997, 134(1), 169–189
  • [16] Hou T.Y., Wu X.-H., A multiscale finite element method for PDEs with oscillatory coefficients, In: Numerical Treatment of Multi-Scale Problems, Kiel, January 24–26, 1997, Notes Numer. Fluid Mech., 70, Friedrich Vieweg & Sohn, Braunschweig, 1999, 58–69
  • [17] Hou T.Y., Wu X.-H., Cai Z., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp., 1999, 68(227), 913–943
  • [18] Hughes T.J.R., The Finite Element Method, Prentice Hall, Englewood Cliffs, 1987
  • [19] Iliev O., Lazarov R., Willems J., Variational multiscale finite element method for flows in highly porous media, Multiscale Model. Simul., 2011, 9(4), 1350–1372
  • [20] Janka A., Algebraic domain decomposition solver for linear elasticity, In: Programs and Algorithms of Numerical Mathematics, Kořenov, June 8–12, 1998, Appl. Math., 1999, 44(6), 435–458
  • [21] Karer E., Subspace Correction Methods for Linear Elasticity, PhD thesis, Universität Linz, 2011
  • [22] Karer E., Kraus J.K., Algebraic multigrid for finite element elasticity equations: determination of nodal dependence via edge-matrices and two-level convergence, Internat. J. Numer. Methods Engrg., 2010, 83(5), 642–670
  • [23] Kolev T., Margenov S., AMLI preconditioning of pure displacement non-conforming elasticity FEM systems, In: Numerical Analysis and its Applications, Rousse, 2000, Lecture Notes in Comput. Sci., 1988, Springer, Berlin, 2001, 482–489
  • [24] Kraus J.K., Algebraic multigrid based on computational molecules, 2: Linear elasticity problems, SIAM J. Sci. Comput., 2008, 30(1), 505–524
  • [25] Kraus J., Margenov S., Robust Algebraic Multilevel Methods and Algorithms, Radon Ser. Comput. Appl. Math., 5, De Gruyter, Berlin, 2009
  • [26] Kraus J.K., Schicho J., Algebraic multigrid based on computational molecules, 1: scalar elliptic problems, Computing, 2006, 77(1), 57–75
  • [27] Mandel J., Brezina M., Vaněk P., Energy optimization of algebraic multigrid bases, Computing, 1999, 62(3), 205–228
  • [28] Millward R., A New Adaptive Multiscale Finite Element Method with Applications to High Contrast Interface Problems, PhD thesis, University of Bath, 2011, available at
  • [29] Saad Y., Iterative Methods for Sparse Linear Systems, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2003
  • [30] Sarkis M., Partition of unity coarse spaces: enhanced versions, discontinuous coefficients and applications to elasticity, In: Domain Decomposition Methods in Science and Engineering, Cocoyoc, January 6–12, 2002, UNAM, México D.F., 2003, 149–158, available at
  • [31] Schulz V., Andrä H., Schmidt K., Robuste Netzgenerierung zur µFE-Analyse mikrostrukturierter Materialien, NAFEMS Magazin, 2007, 7(2), 28–30
  • [32] Smith B.F., Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity, PhD thesis, New York University, 1990
  • [33] 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, Universität Linz, 2011, NuMa-Report #2011-07, available at
  • [34] Toselli A., Widlund O., Domain Decomposition Methods, Algorithms and Theory, Springer Ser. Comput. Math., 34, Springer, Berlin, 2005
  • [35] Vanlent J., Scheichl R., Graham I.G., Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs, Numer. Linear Algebra Appl., 2009, 16(10), 775–799
  • [36] Vaněk P., Acceleration of convergence of a two-level algorithm by smoothing transfer operator, Appl. Math., 1992, 37(4), 265–274
  • [37] Vaněk P., Fast multigrid solver, Appl. Math., 1995, 40(1), 1–20
  • [38] Vaněk P., Brezina M., Tezaur R., Two-grid method for linear elasticity on unstructured meshes, SIAM J. Sci. Comput., 1999, 21(3), 900–923
  • [39] Vassilevski P.S., Multilevel Block Factorization Preconditioners, Springer, New York, 2008
  • [40] Wan W.L., Chan T.F., Smith B., An energy-minimizing interpolation for robust multigrid methods, SIAM J. Sci. Comput., 2000, 21(4), 1632–1649
  • [41] Willems J., Robust multilevel methods for general symmetric positive definite operators, RICAM Institute for Computational and Applied Mathematics, 2012, report #2012-06, available at
  • [42] Xu J., Zikatanov L., On an energy minimizing basis for algebraic multigrid methods, Comput. Vis. Sci., 2004, 7(3–4), 121–127
  • [43] Zhu Y., Sifakis E., Teran J., Brandt A., An efficient multigrid method for the simulation of high resolution elastic solids, ACM Transactions on Graphics, 2010, 29(2), #16
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