Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter October 6, 2016

Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

  • Matthias Maier and Rolf Rannacher EMAIL logo


This paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for a posteriori error estimation with separated quantification of the model error as well as the macroscopic and microscopic discretization errors. The method is derived within a general framework for ‘goal-oriented’ adaptivity, the so-called Dual Weighted Residual (DWR) method. This allows for a systematic a posteriori balancing of multiscale modeling and discretization. The developed method is tested numerically at elliptic diffusion problems for different types of heterogeneous oscillatory coefficients.

MSC 2010: 35J15; 65N12; 65N15; 65N30; 65N50


[1] A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM, SIAM J. Multiscale Modeling Simul., 4 (2005), No. 2, 447-459.10.1137/040607137Search in Google Scholar

[2] A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: a unified framework, In: Multiscale Problems, Series in Contemporary Applied Mathematics, 16 (2011), 280-305.10.1142/9789814366892_0009Search in Google Scholar

[3] A. Abdulle and A. Nonnenmacher, A posteriori error estimate in quantities of interest for the finite element heterogeneous multiscale method, Numer. Meth. PDE, 29 (2013), No. 5, 1629-1656.10.1002/num.21769Search in Google Scholar

[4] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Math. Comp., 83 (2014), 513-536.10.1090/S0025-5718-2013-02758-5Search in Google Scholar

[5] R.A. Adams, SobolevSpaces, PureAppl. Math., Vol. 140, 2003.Search in Google Scholar

[6] G.Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), No. 6, 1482-1518.10.1137/0523084Search in Google Scholar

[7] T. Arbogast and K.J. Boyd, Subgrid upscaling and mixed multiscale finite elements, J. Numer. Anal., 44 (2006), No. 3, 1150-1171.10.1137/050631811Search in Google Scholar

[8] I. Babuska, Homogenization and its application. Mathematical and computational problems. Numerical solution of Partial Differential Equations III, Proc. Third Sympos., Univ. Maryland, 1975, Academic Press, New York, 1976, pp. 363-379.Search in Google Scholar

[9] W. Bangerth, T. Heister, L. Heltai, G. Kanschat, M. Kronbichler, M. Maier, B.Turcksin, and T. D.Young, Thedeal.II Library, version 8.1. arXivpreprint,, 2013.Search in Google Scholar

[10] W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhauser, 2003.10.1007/978-3-0348-7605-6Search in Google Scholar

[11] R. Becker and M. Braack, Multigrid techniques for finite elements on locally refined meshes, Numer. Linear Algebra Appl., 7 (2000), 363-379.10.1002/1099-1506(200009)7:6<363::AID-NLA202>3.0.CO;2-VSearch in Google Scholar

[12] R. Becker and R. Rannacher, Afeed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math., 4 (1996), 237-264.Search in Google Scholar

[13] R. Becker and R. Rannacher, Weighted a posteriori error control in fe methods. In: Proc. ofENUMATH-97, 1998. Lecture at ENUMATH-95, Paris, September 18-22, 1995, pp. 621-637.Search in Google Scholar

[14] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica, 10 (2001), 1-102.10.1017/S0962492901000010Search in Google Scholar

[15] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications, No. 5, North-Holland, 1978.Search in Google Scholar

[16] M. Braack and A. Ern, A posteriori control of modeling errors and discretization errors, Multiscale Modeling Simul., 1 (2003), No. 2, 221-238.10.1137/S1540345902410482Search in Google Scholar

[17] M. Braack and T. Richter, Solutions of 3D Navier-Stokes benchmark problems with adaptive finite element, Computers and Fluids, 35 (2006), 37-392.10.1016/j.compfluid.2005.02.001Search in Google Scholar

[18] S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer, Berlin-Heidelberg-New York, 1994.10.1007/978-1-4757-4338-8Search in Google Scholar

[19] F. Brezzi, Interacting with the subgrid world, Numer. Anal., 1999, pp. 69-82.Search in Google Scholar

[20] W. T. Cardwell Jr. and R. L. Parsons, Average permeability of heterogeneous oil sands, Trans. oftheAIME, 160 (1945), No. 1, 34-42.10.2118/945034-GSearch in Google Scholar

[21] G. F.Carey and J. T.Oden, Finite Elements, Computational Aspects. Vol. III. Prentice-Hall, 1984.Search in Google Scholar

[22] Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2003), 541-576.10.1090/S0025-5718-02-01441-2Search in Google Scholar

[23] P. G. Ciarlet, The Finite Element Method for Elliptic Problems. Classics Appl. Math. Vol. 40, SIAM, Philadelphia, 200210.1137/1.9780898719208Search in Google Scholar

[24] D. Cioranescu and P. Donato, An Introduction to Homogenization. Oxford Lecture Series in Mathematics and Its Applications, Vol. 17. Oxford University Press, 1999.Search in Google Scholar

[25] Ph. Clement, Approximation by finite element functions using local regularization, RAIRO Analyse Numerique9 (1975), 77-84.10.1051/m2an/197509R200771Search in Google Scholar

[26] W. E and B. Engquist, The heterogeneous multiscale methods, Comm. Math. Sci., 1 (2003), No. 1, 87-132.10.4310/CMS.2003.v1.n1.a8Search in Google Scholar

[27] W. E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc., 50 (2003), No. 9, 1062-1070.Search in Google Scholar

[28] W. E, P. Ming, and P.Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18 (2005), 121-156.10.1090/S0894-0347-04-00469-2Search in Google Scholar

[29] Y. Efendiev, V. Ginting, and T. Y. Hou, Multiscale finite element methods for nonlinear problems and their applications, Comm. Math. Sci., 2 (2004), No. 4, 553-589.10.4310/CMS.2004.v2.n4.a2Search in Google Scholar

[30] M. G. D. Geers, V. G. Kouznetsova, and W. A. M. Brekelmans, Multi-scale computational homogenization: Trends and challenges. J. Comp. Appl. Math., 234 (2010), No. 7, 2175-2182.10.1016/ in Google Scholar

[31] P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift, Z. Anal. Anwend., 30 (2011), No. 3, 319-339.10.4171/ZAA/1437Search in Google Scholar

[32] P. Henning, M. Ohlberger, and B. Schweizer, An adaptive multiscale finite element method, Technical Report 05/12-N, FB 10, Universität Münster, 2012.10.1137/120886856Search in Google Scholar

[33] R.Hill, On constitutive macro-variables for heterogeneous solids at finite strain, Proc. Royal Soc., A326 (1972), 131-147.10.1098/rspa.1972.0001Search in Google Scholar

[34] T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comp. Phys., 134 (1997), 169-189.10.1006/jcph.1997.5682Search in Google Scholar

[35] T. J. R. Hughes, G. R. Feijeo, L. Mazzei, and J.-B. Quincy, The variational multiscale method — a paradigm for computational mechanics, Comp. Meth.Appl. Mech. Engrg, 166 (1998), No. 1-2, 3-24.10.1016/S0045-7825(98)00079-6Search in Google Scholar

[36] D. Li, B. Beckner, and A. Kumar, SPE 56554 — a new efficient averaging technique for scaleup of multimillion-cell geologic models, SPE Papers, pp. 495-510, 1999.10.2118/56554-MSSearch in Google Scholar

[37] A.-M. Matache and C. Schwab, Generalized FEM for homogenization problems, Multiscale and Multiresolution Methods, 20 (2002), 197-237.10.1007/978-3-642-56205-1_4Search in Google Scholar

[38] A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems, Math. Modelling Numer. Anal., 36 (2002), No. 4, 536-572.10.1051/m2an:2002025Search in Google Scholar

[39] J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I: Error estimates and adaptive algorithms. J. Comp. Phys., 164 (2000), 22-47.10.1006/jcph.2000.6585Search in Google Scholar

[40] J. T. Oden and K. S. Vemaganti, Adaptive modeling of composite structures: Modeling error estimation. Int. J. Civil Struct. Engrg, 1 (2000), 1-16.Search in Google Scholar

[41] J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. Comp. Meth. Appl. Mech. Engrg, 190 (2001), No. 46-47, 6089-6124.10.1016/S0045-7825(01)00217-1Search in Google Scholar

[42] J. T. Oden, S. Prudhomme, A. Romkes, and P. T. Bauman, Multiscale modeling of physical phenomena: Adaptive control of models. SIAM J. Sci. Comp., 28 (2006), No. 6, 2359-2389.10.1137/050632488Search in Google Scholar

[43] M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Modeling and Simulation, 4 (2005), No. 1, 88-114.10.1137/040605229Search in Google Scholar

[44] T. Richter and T. Wick, Variational localizations of the dual weighted residual estimator, J. Comp. Appl. Math., 271 (2014), 69-85.10.1016/ in Google Scholar

[45] A. Romkes and T. C. Moody, Local goal-oriented estimation of modeling error for multi-scale modeling of heterogeneous elastic materials, Int. J. Comp. Meth. Engrg. Sci. Mech., 4, (2007).10.1080/15502280701375395Search in Google Scholar

[46] L. R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54 (1990), 483-493.10.1090/S0025-5718-1990-1011446-7Search in Google Scholar

[47] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, John Wiley, Teubner, 1996.Search in Google Scholar

[48] H.-J. Vogel, QuantIm, C/C++ library for scientific image processing, version 4.01., 2008.Search in Google Scholar

[49] J. E. Warren and H. S. Price, Flow in heterogeneous porous media, Soc. Petrol. Engineers J., 1 (1961), No. 3, 153-169.10.2118/1579-GSearch in Google Scholar

Received: 2014-10-28
Revised: 2015-4-15
Accepted: 2015-7-2
Published Online: 2016-10-6
Published in Print: 2016-10-1

© 2016 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.9.2023 from
Scroll to top button