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December  2010, 5(4): 711-744. doi: 10.3934/nhm.2010.5.711

The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift

1. 

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

Received  March 2010 Revised  May 2010 Published  November 2010

This contribution is concerned with the formulation of a heterogeneous multiscale finite elements method (HMM) for solving linear advection-diffusion problems with rapidly oscillating coefficient functions and a large expected drift. We show that, in the case of periodic coefficient functions, this approach is equivalent to a discretization of the two-scale homogenized equation by means of a Discontinuous Galerkin Time Stepping Method with quadrature. We then derive an optimal order a-priori error estimate for this version of the HMM and finally provide numerical experiments to validate the method.
Citation: Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711
References:
[1]

A. Abdulle, Multiscale methods for advection-diffusion problems,, Discrete Contin. Dyn. Syst., suppl (2005), 11. Google Scholar

[2]

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

[3]

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

[4]

A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces,, Multiscale Model. Simul., 3 (2004), 195. doi: 10.1137/030600771. Google Scholar

[5]

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. Google Scholar

[6]

G. Allaire and A.-L. Raphael, "Homogénéisation d'un Modèle de Convection-Diffusion Avec Chimie/Adsorption en Milieu Poreux," (French),, Rapport Interne, n. 604 (2006). Google Scholar

[7]

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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. 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, 44 (2005), 89. 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 and T. 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. Google Scholar

[15]

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. Google Scholar

[16]

P. Henning and M. Ohlberger, A-posteriori error estimate for a heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift,, Preprint, N-09/09 (2009). Google Scholar

[17]

P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift,, submitted to: ZAA, (2010). Google Scholar

[18]

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

[19]

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

[20]

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. Google Scholar

[21]

T. Y. Hou, X.-H. Wu and C. Zhiqiang, 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. Google Scholar

[22]

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. Google Scholar

[23]

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. Google Scholar

[24]

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

[25]

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. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

C. Schwab and A.-M. Matache, Generalized FEM for homogenization problems,, in, 20 (2002), 197. Google Scholar

[31]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997). Google Scholar

[32]

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), 46. doi: 10.1016/S0045-7825(01)00217-1. Google Scholar

show all references

References:
[1]

A. Abdulle, Multiscale methods for advection-diffusion problems,, Discrete Contin. Dyn. Syst., suppl (2005), 11. Google Scholar

[2]

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

[3]

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

[4]

A. Abdulle and C. Schwab, Heterogeneous multiscale FEM for diffusion problems on rough surfaces,, Multiscale Model. Simul., 3 (2004), 195. doi: 10.1137/030600771. Google Scholar

[5]

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. Google Scholar

[6]

G. Allaire and A.-L. Raphael, "Homogénéisation d'un Modèle de Convection-Diffusion Avec Chimie/Adsorption en Milieu Poreux," (French),, Rapport Interne, n. 604 (2006). Google Scholar

[7]

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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. 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, 44 (2005), 89. 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 and T. 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. Google Scholar

[15]

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. Google Scholar

[16]

P. Henning and M. Ohlberger, A-posteriori error estimate for a heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift,, Preprint, N-09/09 (2009). Google Scholar

[17]

P. Henning and M. Ohlberger, A note on homogenization of advection-diffusion problems with large expected drift,, submitted to: ZAA, (2010). Google Scholar

[18]

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

[19]

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

[20]

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. Google Scholar

[21]

T. Y. Hou, X.-H. Wu and C. Zhiqiang, 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. Google Scholar

[22]

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. Google Scholar

[23]

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. Google Scholar

[24]

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

[25]

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. Google Scholar

[26]

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

[27]

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

[28]

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

[29]

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

[30]

C. Schwab and A.-M. Matache, Generalized FEM for homogenization problems,, in, 20 (2002), 197. Google Scholar

[31]

V. Thomée, "Galerkin Finite Element Methods for Parabolic Problems,", Springer Series in Computational Mathematics, 25 (1997). Google Scholar

[32]

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), 46. doi: 10.1016/S0045-7825(01)00217-1. Google Scholar

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