October  2016, 9(5): 1393-1420. doi: 10.3934/dcdss.2016056

A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift

1. 

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm

2. 

Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

Received  January 2015 Revised  March 2015 Published  October 2016

In this contribution we address a-posteriori error estimation in $L^\infty(L^2)$ for a heterogeneous multiscale finite element approximation of time-dependent advection-diffusion problems with rapidly oscillating coefficient functions and with a large expected drift. Based on the error estimate, we derive an algorithm for an adaptive mesh refinement. The estimate and the algorithm are validated in numerical experiments, showing applicability and good results even for heterogeneous microstructures.
Citation: Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
References:
[1]

A. Abdulle, Multiscale methods for advection-diffusion problems,, Discrete and Continuous Dynamical Systems Series A, 5 (2005), 11.

[2]

A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM,, Multiscale Model. Simul., 4 (2005), 447. doi: 10.1137/040607137.

[3]

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

[4]

A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems,, J. Comput. Phys., 231 (2012), 7014. doi: 10.1016/j.jcp.2012.02.019.

[5]

A. Abdulle and Y. Bai, Adaptive reduced basis finite element heterogeneous multiscale method,, Comput. Methods Appl. Mech. Engrg., 257 (2013), 203. doi: 10.1016/j.cma.2013.01.002.

[6]

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.

[7]

A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. doi: 10.1017/S0962492912000025.

[8]

A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the wave equation,, Multiscale Model. Simul., 9 (2011), 766. doi: 10.1137/100800488.

[9]

A. Abdulle and M. E. Huber, Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales,, Numer. Math., 126 (2014), 589. doi: 10.1007/s00211-013-0578-9.

[10]

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.

[11]

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.

[12]

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

[13]

A. Abdulle and G. Vilmart, The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods,, C. R. Math. Acad. Sci. Paris, 349 (2011), 1041. doi: 10.1016/j.crma.2011.09.005.

[14]

G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential,, ESAIM Control Optim. Calc. Var., 13 (2007), 735. doi: 10.1051/cocv:2007030.

[15]

G. Allaire and A.-L. Raphael, Homogénéisation d'un modèle de convection-diffusion avec chimie/adsorption en milieu poreux,, Rapport Interne, (2006).

[16]

G. Allaire and A.-L. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium,, C. R. Math. Acad. Sci. Paris, 344 (2007), 523. doi: 10.1016/j.crma.2007.03.008.

[17]

T. Arbogast, G. Pencheva, M. F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method,, Multiscale Model. Simul., 6 (2007), 319. doi: 10.1137/060662587.

[18]

A. Bourlioux and A. J. Majda, An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion,, Combust. Theory Model., 4 (2000), 189. doi: 10.1088/1364-7830/4/2/307.

[19]

R. Du and P. B. Ming, Convergence of the heterogeneous multiscale finite element method for elliptic problems with nonsmooth microstructures,, Multiscale Model. Simul., 8 (2010), 1770. doi: 10.1137/090780754.

[20]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87.

[21]

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

[22]

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.

[23]

W. E, P. B. Ming and P. W. 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.

[24]

Y. Efendiev and T. Y. Hou, Multiscale finite element methods for porous media flows and their applications,, Appl. Numer. Math., 57 (2007), 577. doi: 10.1016/j.apnum.2006.07.009.

[25]

Y. R. Efendiev, T. Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method,, SIAM J. Numer. Anal., 37 (2000), 888. doi: 10.1137/S0036142997330329.

[26]

B. Engquist, H. Holst and O. Runborg, Multi-scale methods for wave propagation in heterogeneous media,, Commun. Math. Sci., 9 (2011), 33.

[27]

B. Engquist, H. Holst and O. Runborg, Multiscale methods for wave propagation in heterogeneous media over long time,, In Numerical Analysis of Multiscale Computations, (2012), 167. doi: 10.1007/978-3-642-21943-6_8.

[28]

K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43. doi: 10.1137/0728003.

[29]

K. Eriksson, C. Johnson and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups,, SIAM J. Numer. Anal., 35 (1998), 1315. doi: 10.1137/S0036142996310216.

[30]

F. R. F. Nataf and E. de Sturler, Optimal interface conditions for domain decomposition methods,, CMAP (Ecole Polytechnique), (1994).

[31]

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.

[32]

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

[33]

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.

[34]

V. Gravemeier and W. A. Wall, A 'divide-and-conquer' spatial and temporal multiscale method for transient convection-diffusion-reaction equations,, Internat. J. Numer. Methods Fluids, 54 (2007), 779. doi: 10.1002/fld.1465.

[35]

L. Halpern, Artificial boundary conditions for the linear advection diffusion equation,, Math. Comp., 46 (1986), 425. doi: 10.2307/2007985.

[36]

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.

[37]

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.

[38]

P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift,, Z. Anal. Anwend., 30 (2011), 319. doi: 10.4171/ZAA/1437.

[39]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 119. doi: 10.3934/dcdss.2015.8.119.

[40]

V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul., 3 (): 168. doi: 10.1137/030601077.

[41]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, J. Comput. Phys., 134 (1997), 169. doi: 10.1006/jcph.1997.5682.

[42]

T. Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,, Math. Comp., 68 (1999), 913. doi: 10.1090/S0025-5718-99-01077-7.

[43]

T. Y. Hou, X.-H. Wu and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation,, Commun. Math. Sci., 2 (2004), 185.

[44]

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.

[45]

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.

[46]

L. Jiang, Y. Efendiev and V. Ginting, Multiscale methods for parabolic equations with continuum spatial scales,, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 833. doi: 10.3934/dcdsb.2007.8.833.

[47]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). doi: 10.1007/978-3-642-84659-5.

[48]

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.

[49]

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.

[50]

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.

[51]

R. Li, P. B. Ming and F. Tang, An efficient high order heterogeneous multiscale method for elliptic problems,, Multiscale Model. Simul., 10 (2012), 259. doi: 10.1137/110836626.

[52]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems,, Math. Comp., 83 (2014), 2583. doi: 10.1090/S0025-5718-2014-02868-8.

[53]

E. Marušić Paloka and A. L. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection,, J. London Math. Soc. (2), 72 (2005), 391. doi: 10.1112/S0024610705006824.

[54]

A.-M. Matache, Sparse two-scale FEM for homogenization problems,, In Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 17 (2002), 659. doi: 10.1023/A:1015187000835.

[55]

A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems,, M2AN Math. Model. Numer. Anal., 36 (2002), 537. doi: 10.1051/m2an:2002025.

[56]

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.

[57]

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.

[58]

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.

[59]

C. Schwab and A.-M. Matache, Generalized FEM for homogenization problems,, In Multiscale and multiresolution methods, (2002), 197. doi: 10.1007/978-3-642-56205-1_4.

[60]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, volume 25 of Springer Series in Computational Mathematics,, Springer-Verlag, (1997). doi: 10.1007/978-3-662-03359-3.

show all references

References:
[1]

A. Abdulle, Multiscale methods for advection-diffusion problems,, Discrete and Continuous Dynamical Systems Series A, 5 (2005), 11.

[2]

A. Abdulle, On a priori error analysis of fully discrete heterogeneous multiscale FEM,, Multiscale Model. Simul., 4 (2005), 447. doi: 10.1137/040607137.

[3]

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

[4]

A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems,, J. Comput. Phys., 231 (2012), 7014. doi: 10.1016/j.jcp.2012.02.019.

[5]

A. Abdulle and Y. Bai, Adaptive reduced basis finite element heterogeneous multiscale method,, Comput. Methods Appl. Mech. Engrg., 257 (2013), 203. doi: 10.1016/j.cma.2013.01.002.

[6]

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.

[7]

A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. doi: 10.1017/S0962492912000025.

[8]

A. Abdulle and M. J. Grote, Finite element heterogeneous multiscale method for the wave equation,, Multiscale Model. Simul., 9 (2011), 766. doi: 10.1137/100800488.

[9]

A. Abdulle and M. E. Huber, Discontinuous Galerkin finite element heterogeneous multiscale method for advection-diffusion problems with multiple scales,, Numer. Math., 126 (2014), 589. doi: 10.1007/s00211-013-0578-9.

[10]

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.

[11]

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.

[12]

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

[13]

A. Abdulle and G. Vilmart, The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods,, C. R. Math. Acad. Sci. Paris, 349 (2011), 1041. doi: 10.1016/j.crma.2011.09.005.

[14]

G. Allaire and R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential,, ESAIM Control Optim. Calc. Var., 13 (2007), 735. doi: 10.1051/cocv:2007030.

[15]

G. Allaire and A.-L. Raphael, Homogénéisation d'un modèle de convection-diffusion avec chimie/adsorption en milieu poreux,, Rapport Interne, (2006).

[16]

G. Allaire and A.-L. Raphael, Homogenization of a convection-diffusion model with reaction in a porous medium,, C. R. Math. Acad. Sci. Paris, 344 (2007), 523. doi: 10.1016/j.crma.2007.03.008.

[17]

T. Arbogast, G. Pencheva, M. F. Wheeler and I. Yotov, A multiscale mortar mixed finite element method,, Multiscale Model. Simul., 6 (2007), 319. doi: 10.1137/060662587.

[18]

A. Bourlioux and A. J. Majda, An elementary model for the validation of flamelet approximations in non-premixed turbulent combustion,, Combust. Theory Model., 4 (2000), 189. doi: 10.1088/1364-7830/4/2/307.

[19]

R. Du and P. B. Ming, Convergence of the heterogeneous multiscale finite element method for elliptic problems with nonsmooth microstructures,, Multiscale Model. Simul., 8 (2010), 1770. doi: 10.1137/090780754.

[20]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87.

[21]

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

[22]

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.

[23]

W. E, P. B. Ming and P. W. 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.

[24]

Y. Efendiev and T. Y. Hou, Multiscale finite element methods for porous media flows and their applications,, Appl. Numer. Math., 57 (2007), 577. doi: 10.1016/j.apnum.2006.07.009.

[25]

Y. R. Efendiev, T. Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method,, SIAM J. Numer. Anal., 37 (2000), 888. doi: 10.1137/S0036142997330329.

[26]

B. Engquist, H. Holst and O. Runborg, Multi-scale methods for wave propagation in heterogeneous media,, Commun. Math. Sci., 9 (2011), 33.

[27]

B. Engquist, H. Holst and O. Runborg, Multiscale methods for wave propagation in heterogeneous media over long time,, In Numerical Analysis of Multiscale Computations, (2012), 167. doi: 10.1007/978-3-642-21943-6_8.

[28]

K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. I. A linear model problem,, SIAM J. Numer. Anal., 28 (1991), 43. doi: 10.1137/0728003.

[29]

K. Eriksson, C. Johnson and S. Larsson, Adaptive finite element methods for parabolic problems. VI. Analytic semigroups,, SIAM J. Numer. Anal., 35 (1998), 1315. doi: 10.1137/S0036142996310216.

[30]

F. R. F. Nataf and E. de Sturler, Optimal interface conditions for domain decomposition methods,, CMAP (Ecole Polytechnique), (1994).

[31]

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.

[32]

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

[33]

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.

[34]

V. Gravemeier and W. A. Wall, A 'divide-and-conquer' spatial and temporal multiscale method for transient convection-diffusion-reaction equations,, Internat. J. Numer. Methods Fluids, 54 (2007), 779. doi: 10.1002/fld.1465.

[35]

L. Halpern, Artificial boundary conditions for the linear advection diffusion equation,, Math. Comp., 46 (1986), 425. doi: 10.2307/2007985.

[36]

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.

[37]

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.

[38]

P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift,, Z. Anal. Anwend., 30 (2011), 319. doi: 10.4171/ZAA/1437.

[39]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems,, Discrete Contin. Dyn. Syst. Ser. S, 8 (2015), 119. doi: 10.3934/dcdss.2015.8.119.

[40]

V. H. Hoang and C. Schwab, High-dimensional finite elements for elliptic problems with multiple scales,, Multiscale Model. Simul., 3 (): 168. doi: 10.1137/030601077.

[41]

T. Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, J. Comput. Phys., 134 (1997), 169. doi: 10.1006/jcph.1997.5682.

[42]

T. Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,, Math. Comp., 68 (1999), 913. doi: 10.1090/S0025-5718-99-01077-7.

[43]

T. Y. Hou, X.-H. Wu and Y. Zhang, Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation,, Commun. Math. Sci., 2 (2004), 185.

[44]

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.

[45]

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.

[46]

L. Jiang, Y. Efendiev and V. Ginting, Multiscale methods for parabolic equations with continuum spatial scales,, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 833. doi: 10.3934/dcdsb.2007.8.833.

[47]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). doi: 10.1007/978-3-642-84659-5.

[48]

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.

[49]

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.

[50]

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.

[51]

R. Li, P. B. Ming and F. Tang, An efficient high order heterogeneous multiscale method for elliptic problems,, Multiscale Model. Simul., 10 (2012), 259. doi: 10.1137/110836626.

[52]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems,, Math. Comp., 83 (2014), 2583. doi: 10.1090/S0025-5718-2014-02868-8.

[53]

E. Marušić Paloka and A. L. Piatnitski, Homogenization of a nonlinear convection-diffusion equation with rapidly oscillating coefficients and strong convection,, J. London Math. Soc. (2), 72 (2005), 391. doi: 10.1112/S0024610705006824.

[54]

A.-M. Matache, Sparse two-scale FEM for homogenization problems,, In Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01) (Uppsala), 17 (2002), 659. doi: 10.1023/A:1015187000835.

[55]

A.-M. Matache and C. Schwab, Two-scale FEM for homogenization problems,, M2AN Math. Model. Numer. Anal., 36 (2002), 537. doi: 10.1051/m2an:2002025.

[56]

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.

[57]

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.

[58]

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.

[59]

C. Schwab and A.-M. Matache, Generalized FEM for homogenization problems,, In Multiscale and multiresolution methods, (2002), 197. doi: 10.1007/978-3-642-56205-1_4.

[60]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, volume 25 of Springer Series in Computational Mathematics,, Springer-Verlag, (1997). doi: 10.1007/978-3-662-03359-3.

[1]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

[2]

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