• Previous Article
    Classical solutions and feedback stabilization for the gas flow in a sequence of pipes
  • NHM Home
  • This Issue
  • Next Article
    Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients
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.

[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 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.

[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.

[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.

[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).

[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.

[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.

[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.

[10]

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

[11]

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

[12]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in, 44 (2005), 89.

[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.

[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.

[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.

[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).

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

show all references

References:
[1]

A. Abdulle, Multiscale methods for advection-diffusion problems,, Discrete Contin. Dyn. Syst., suppl (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 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.

[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.

[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.

[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).

[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.

[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.

[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.

[10]

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

[11]

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

[12]

W. E and B. Engquist, The heterogeneous multi-scale method for homogenization problems,, in, 44 (2005), 89.

[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.

[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.

[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.

[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).

[17]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[30]

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

[31]

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

[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.

[1]

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

[2]

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

[3]

Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641

[4]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200

[5]

Tao Lin, Yanping Lin, Weiwei Sun. Error estimation of a class of quadratic immersed finite element methods for elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 807-823. doi: 10.3934/dcdsb.2007.7.807

[6]

Huan-Zhen Chen, Zhao-Jie Zhou, Hong Wang, Hong-Ying Man. An optimal-order error estimate for a family of characteristic-mixed methods to transient convection-diffusion problems. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 325-341. doi: 10.3934/dcdsb.2011.15.325

[7]

Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261

[8]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[9]

Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1

[10]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161

[11]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017

[12]

Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663

[13]

François Alouges. A new finite element scheme for Landau-Lifchitz equations. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 187-196. doi: 10.3934/dcdss.2008.1.187

[14]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[15]

Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295

[16]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927

[17]

Qun Lin, Hehu Xie. Recent results on lower bounds of eigenvalue problems by nonconforming finite element methods. Inverse Problems & Imaging, 2013, 7 (3) : 795-811. doi: 10.3934/ipi.2013.7.795

[18]

Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689

[19]

Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034

[20]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]