February  2015, 8(1): 119-150. doi: 10.3934/dcdss.2015.8.119

Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems

1. 

Division of Scientific Computing, Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden

2. 

Institut für Numerische und Angewandte Mathematik, Fachbereich Mathematik und Informatik der Universität Münster, Einsteinstrasse 62, 48149 Münster

Received  February 2012 Revised  April 2013 Published  July 2014

In this work we introduce and analyse a new adaptive Petrov-Galerkin heterogeneous multiscale finite element method (HMM) for monotone elliptic operators with rapid oscillations. In a general heterogeneous setting we prove convergence of the HMM approximations to the solution of a macroscopic limit equation. The major new contribution of this work is an a-posteriori error estimate for the $L^2$-error between the HMM approximation and the solution of the macroscopic limit equation. The a posteriori error estimate is obtained in a general heterogeneous setting with scale separation without assuming periodicity or stochastic ergodicity. The applicability of the method and the usage of the a posteriori error estimate for adaptive local mesh refinement is demonstrated in numerical experiments. The experimental results underline the applicability of the a posteriori error estimate in non-periodic homogenization settings.
Citation: Patrick Henning, Mario Ohlberger. Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 119-150. doi: 10.3934/dcdss.2015.8.119
References:
[1]

A. Abdulle, The finite element heterogeneous multiscale method: A computational strategy for multiscale PDEs,, in Multiple scales problems in biomathematics, (2009), 133. Google Scholar

[2]

A. Abdulle and A. Nonnenmacher, Adaptive finite element heterogeneous multiscale method for homogenization problems,, Comput. Methods Appl. Mech. Engrg., 200 (2011), 2710. doi: 10.1016/j.cma.2010.06.012. Google Scholar

[3]

A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems,, submitted for publication, (2012). Google Scholar

[4]

A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems,, J. Comput. Phys., 191 (2003), 18. doi: 10.1016/S0021-9991(03)00303-6. Google Scholar

[5]

A. Abdulle and A. Nonnenmacher, A posteriori error analysis of the heterogeneous multiscale method for homogenization problems,, C. R. Math. Acad. Sci. Paris, 347 (2009), 1081. doi: 10.1016/j.crma.2009.07.004. Google Scholar

[6]

A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces,, Multiscale Model. Simul., 3 (): 195. doi: 10.1137/030600771. Google Scholar

[7]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[8]

H. W. Alt, $^4$ Lineare Funktionalanalysis,, Springer, (2002). Google Scholar

[9]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Studies in Mathematics and its Applications, (1978). Google Scholar

[10]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. doi: 10.4310/CMS.2003.v1.n1.a8. Google Scholar

[11]

W. E and B. Engquist, Multiscale modeling and computation,, Notices Amer. Math. Soc., 50 (2003), 1062. Google Scholar

[12]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in Multiscale Methods in Science and Engineering, (2005), 89. doi: 10.1007/3-540-26444-2_4. Google Scholar

[13]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[14]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. doi: 10.4310/CMS.2004.v2.n4.a2. Google Scholar

[15]

Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods. Theory and Applications,, Surveys and Tutorials in the Applied Mathematical Sciences, (2009). Google Scholar

[16]

A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies,, Multiscale Model. Simul., 5 (2006), 996. doi: 10.1137/060649112. Google Scholar

[17]

A. Gloria, An analytical framework for numerical homogenization. {II}. Windowing and oversampling,, Multiscale Model. Simul., 7 (2008), 274. doi: 10.1137/070683143. Google Scholar

[18]

A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients,, Math. Models Methods Appl. Sci., 21 (2011), 1601. doi: 10.1142/S0218202511005507. Google Scholar

[19]

A. Gloria, Numerical homogenization: Survey, new results, and perspectives,, in Mathematical and Numerical Approaches for Multiscale Problem., (2012), 50. doi: 10.1051/proc/201237002. Google Scholar

[20]

P. Grisvard, Singularities in Boundary Value Problems,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1992). Google Scholar

[21]

P. Henning, M. Ohlberger and B. Schweizer, An adaptive multiscale finite element method,, University of Münster, (2012). Google Scholar

[22]

P. Henning, Heterogeneous Multiscale Finite Element Methods for Advection-Diffusion and Nonlinear Elliptic Multiscale Problems,, Münster: Univ. Münster, (2011). Google Scholar

[23]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math., 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[24]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift,, Netw. Heterog. Media, 5 (2010), 711. doi: 10.3934/nhm.2010.5.711. Google Scholar

[25]

P. Henning and M. Ohlberger, On the implementation of a heterogeneous multiscale finite element method for nonlinear elliptic problems,, in Advances in DUNE. Proceedings of the DUNE User Meeting, (2012), 143. Google Scholar

[26]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[27]

T. J. R. Hughes, G. R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[28]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Duality techniques for elliptic problems,, in Multiscale Methods in Science and Engineering, (2005), 181. doi: 10.1007/3-540-26444-2_9. Google Scholar

[29]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[30]

M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems,, Comm. Numer. Methods Engrg., 25 (2009), 65. doi: 10.1002/cnm.1106. Google Scholar

[31]

J. L. Lions, D. Lukkassen, L. E. Persson and P. Wall, Reiterated homogenization of nonlinear monotone operators,, Chinese Ann. Math. Ser. B, 22 (2001), 1. doi: 10.1142/S0252959901000024. Google Scholar

[32]

D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35. Google Scholar

[33]

P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for parabolic homogenization problems,, Math. Comp., 76 (2007), 153. doi: 10.1090/S0025-5718-06-01909-0. Google Scholar

[34]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. doi: 10.1137/070693230. Google Scholar

[35]

J. M. Nordbotten, Variational and heterogeneous multiscale methods for non-linear problems,, in Proc. of ENUMATH 2009, (2009). Google Scholar

[36]

J. T. Oden and K. S. Vemaganti, Adaptive modeling of composite structures: Modeling error estimation,, Int. J. Comp. Civil Str. Engrg., 1 (2000), 1. Google Scholar

[37]

J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms,, J. Comput. Phys., 164 (2000), 22. doi: 10.1006/jcph.2000.6585. Google Scholar

[38]

M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems,, Multiscale Model. Simul., 4 (2005), 88. doi: 10.1137/040605229. Google Scholar

[39]

M. Růžička, Nichtlineare Funktionalanalysis,, Springer-Verlag, (2004). Google Scholar

[40]

K. S. Vemaganti and J. T. Oden, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. II. A computational environment for adaptive modeling of heterogeneous elastic solids,, Comput. Methods Appl. Mech. Engrg., 190 (2001), 6089. doi: 10.1016/S0045-7825(01)00217-1. Google Scholar

[41]

P. Wall, Some homogenization and corrector results for nonlinear monotone operators,, J. Nonlinear Math. Phys., 5 (1998), 331. doi: 10.2991/jnmp.1998.5.3.7. Google Scholar

[42]

T. I. Zohdi, J. T. Oden and G. J. Rodin, Hierarchical modeling of heterogeneous bodies,, Comput. Methods Appl. Mech. Engrg., 138 (1996), 273. doi: 10.1016/S0045-7825(96)01106-1. Google Scholar

show all references

References:
[1]

A. Abdulle, The finite element heterogeneous multiscale method: A computational strategy for multiscale PDEs,, in Multiple scales problems in biomathematics, (2009), 133. Google Scholar

[2]

A. Abdulle and A. Nonnenmacher, Adaptive finite element heterogeneous multiscale method for homogenization problems,, Comput. Methods Appl. Mech. Engrg., 200 (2011), 2710. doi: 10.1016/j.cma.2010.06.012. Google Scholar

[3]

A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems,, submitted for publication, (2012). Google Scholar

[4]

A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems,, J. Comput. Phys., 191 (2003), 18. doi: 10.1016/S0021-9991(03)00303-6. Google Scholar

[5]

A. Abdulle and A. Nonnenmacher, A posteriori error analysis of the heterogeneous multiscale method for homogenization problems,, C. R. Math. Acad. Sci. Paris, 347 (2009), 1081. doi: 10.1016/j.crma.2009.07.004. Google Scholar

[6]

A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces,, Multiscale Model. Simul., 3 (): 195. doi: 10.1137/030600771. Google Scholar

[7]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar

[8]

H. W. Alt, $^4$ Lineare Funktionalanalysis,, Springer, (2002). Google Scholar

[9]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems,, Studies in Mathematics and its Applications, (1978). Google Scholar

[10]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. doi: 10.4310/CMS.2003.v1.n1.a8. Google Scholar

[11]

W. E and B. Engquist, Multiscale modeling and computation,, Notices Amer. Math. Soc., 50 (2003), 1062. Google Scholar

[12]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in Multiscale Methods in Science and Engineering, (2005), 89. doi: 10.1007/3-540-26444-2_4. Google Scholar

[13]

W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems,, J. Amer. Math. Soc., 18 (2005), 121. doi: 10.1090/S0894-0347-04-00469-2. Google Scholar

[14]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. doi: 10.4310/CMS.2004.v2.n4.a2. Google Scholar

[15]

Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods. Theory and Applications,, Surveys and Tutorials in the Applied Mathematical Sciences, (2009). Google Scholar

[16]

A. Gloria, An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies,, Multiscale Model. Simul., 5 (2006), 996. doi: 10.1137/060649112. Google Scholar

[17]

A. Gloria, An analytical framework for numerical homogenization. {II}. Windowing and oversampling,, Multiscale Model. Simul., 7 (2008), 274. doi: 10.1137/070683143. Google Scholar

[18]

A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients,, Math. Models Methods Appl. Sci., 21 (2011), 1601. doi: 10.1142/S0218202511005507. Google Scholar

[19]

A. Gloria, Numerical homogenization: Survey, new results, and perspectives,, in Mathematical and Numerical Approaches for Multiscale Problem., (2012), 50. doi: 10.1051/proc/201237002. Google Scholar

[20]

P. Grisvard, Singularities in Boundary Value Problems,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], (1992). Google Scholar

[21]

P. Henning, M. Ohlberger and B. Schweizer, An adaptive multiscale finite element method,, University of Münster, (2012). Google Scholar

[22]

P. Henning, Heterogeneous Multiscale Finite Element Methods for Advection-Diffusion and Nonlinear Elliptic Multiscale Problems,, Münster: Univ. Münster, (2011). Google Scholar

[23]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math., 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[24]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift,, Netw. Heterog. Media, 5 (2010), 711. doi: 10.3934/nhm.2010.5.711. Google Scholar

[25]

P. Henning and M. Ohlberger, On the implementation of a heterogeneous multiscale finite element method for nonlinear elliptic problems,, in Advances in DUNE. Proceedings of the DUNE User Meeting, (2012), 143. Google Scholar

[26]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[27]

T. J. R. Hughes, G. R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[28]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Duality techniques for elliptic problems,, in Multiscale Methods in Science and Engineering, (2005), 181. doi: 10.1007/3-540-26444-2_9. Google Scholar

[29]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[30]

M. G. Larson and A. Målqvist, An adaptive variational multiscale method for convection-diffusion problems,, Comm. Numer. Methods Engrg., 25 (2009), 65. doi: 10.1002/cnm.1106. Google Scholar

[31]

J. L. Lions, D. Lukkassen, L. E. Persson and P. Wall, Reiterated homogenization of nonlinear monotone operators,, Chinese Ann. Math. Ser. B, 22 (2001), 1. doi: 10.1142/S0252959901000024. Google Scholar

[32]

D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35. Google Scholar

[33]

P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for parabolic homogenization problems,, Math. Comp., 76 (2007), 153. doi: 10.1090/S0025-5718-06-01909-0. Google Scholar

[34]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. doi: 10.1137/070693230. Google Scholar

[35]

J. M. Nordbotten, Variational and heterogeneous multiscale methods for non-linear problems,, in Proc. of ENUMATH 2009, (2009). Google Scholar

[36]

J. T. Oden and K. S. Vemaganti, Adaptive modeling of composite structures: Modeling error estimation,, Int. J. Comp. Civil Str. Engrg., 1 (2000), 1. Google Scholar

[37]

J. T. Oden and K. S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms,, J. Comput. Phys., 164 (2000), 22. doi: 10.1006/jcph.2000.6585. Google Scholar

[38]

M. Ohlberger, A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems,, Multiscale Model. Simul., 4 (2005), 88. doi: 10.1137/040605229. Google Scholar

[39]

M. Růžička, Nichtlineare Funktionalanalysis,, Springer-Verlag, (2004). Google Scholar

[40]

K. S. Vemaganti and J. T. Oden, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. II. A computational environment for adaptive modeling of heterogeneous elastic solids,, Comput. Methods Appl. Mech. Engrg., 190 (2001), 6089. doi: 10.1016/S0045-7825(01)00217-1. Google Scholar

[41]

P. Wall, Some homogenization and corrector results for nonlinear monotone operators,, J. Nonlinear Math. Phys., 5 (1998), 331. doi: 10.2991/jnmp.1998.5.3.7. Google Scholar

[42]

T. I. Zohdi, J. T. Oden and G. J. Rodin, Hierarchical modeling of heterogeneous bodies,, Comput. Methods Appl. Mech. Engrg., 138 (1996), 273. doi: 10.1016/S0045-7825(96)01106-1. Google Scholar

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