Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Analysis of the heterogeneous multiscale method for parabolic homogenization problems

Authors: Pingbing Ming and Pingwen Zhang
Journal: Math. Comp. 76 (2007), 153-177
MSC (2000): Primary 65N30, 35K05, 65N15
Published electronically: October 10, 2006
MathSciNet review: 2261016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

References [Enhancements On Off] (What's this?)

  • 1. A. Abdulle and W. E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys. 191 (2003), 18-39. MR 2008485 (2004i:65071)
  • 2. G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Reisz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), 691-727. MR 1178564 (93j:22006)
  • 3. N. André and M. Chipot, Uniqueness and nonuniqueness for the approximation of quasiliear elliptic equations, SIAM J. Numer. Anal. 33 (1996), 1981-1994. MR 1411859 (98k:65064)
  • 4. M. Artola and G. Duvaut, Un résultat d'homogénéisation pour une classe de problémes de diffusion non stationnaires, Ann. Fac. Sci. Toulouse. (5) IV (1982), 1-28. MR 0673637 (84j:35020)
  • 5. N. Bakhvalov and G. Panasenko, Homogenisation: Averaging Processes in Periodic Media, Mathematical Problems in the Mechanics of Materials, Kluwer Academic Publishers, 1989. MR 1112788 (92d:73002)
  • 6. A. Bensoussan, J. L. Lions and G. C. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. MR 0503330 (82h:35001)
  • 7. L. Boccardo and F. Murat, Homogénéisation de problémes quasi-linéaires, Publ. IRMA, Lille. 3 (1981), no. 7, 13-51. MR 0766874 (86a:35022)
  • 8. S. Brahim-Otsmane, G. A. Francfort and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl. 71 (1992), 197-231. MR 1172450 (93d:35012)
  • 9. S. Q. Chen, W. E and C-W. Shu, The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems, Multiscale Model. Simul. 3 (2005), 871-894. MR 2164241 (2006h:65142)
  • 10. P. G. Ciarlet, The Finite Element Method for the Elliptic Problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • 11. F. Colombini and S. Spagnolo, Sur la convergence de solutions d'équations paraboliques, J. Math. Pures Appl. 56 (1977), 263-305. MR 0603300 (58:29248)
  • 12. A. Dall'aglio and F. Murat, A corrector result for H-converging parabolic problems with time-dependent coefficients, Ann. Scuola Norm. Sup. Pisa C1. Sci. XXV (1997), 329-373. MR 1655521 (99m:35023)
  • 13. J. Douglas and T. Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575-626. MR 0277126 (43:2863)
  • 14. W. E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci. 1 (2003), 423-436. MR 2069938 (2005f:65082)
  • 15. W. E and B. Engquist, The heterogeneous multi-scale methods, Commun. Math. Sci. 1 (2003), 87-132. MR 1979846 (2004b:35019)
  • 16. W. E and B. Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc. 50 (2003), 1062-1070. MR 2002752 (2004m:65163)
  • 17. 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), 121-156. MR 2114818 (2005k:65246)
  • 18. L. C. Evans, Partial Differential Equations, American Mathmatical Society, Providence, Rhode Island: AMS, 1998. MR 1625845 (99e:35001)
  • 19. J. Garcia-Azorero, C. E. Gutierrez and I. Peral, Homogenization of quasilinear parabolic equations in periodic media, Comm. Partial Differential Equations. 28 (2003), 1887-1910. MR 2015406 (2004i:35028)
  • 20. O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematics Monographys, Vol. 25, Providence, Rhode Island: AMS, 1968. MR 1195131 (93k:35025)
  • 21. O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992. MR 1195131 (93k:35025)
  • 22. S. Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all'equazione del calore, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 657-699. MR 0225015 (37:612)
  • 23. S. Spagnolo, Sulla convergenza di soluzioni di equationi paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 571-597. MR 0240443 (39:1791)
  • 24. S. Spagnolo, Convergence of parabolic equations, Boll. Un. Mat. Ital. B (5) 14 (1977), 547-568. MR 0460889 (57:880)
  • 25. L. Tartar, H-Convergence, Course Peccot, Collége de France, March 1977. Partially written by F. Murat. Séminaire d'Analyse Fonctionnelle et Numérique de l'Université d'Alger, 1977-1978. English Translation: F. Murat and L. Tartar: H-Convergence, in Topics in the Mathematical Modeling of Composite Materials, A. Cherkaev and R. Kohn, eds., Birkhäuser, Boston, 1997, pp. 21-43. MR 1493039
  • 26. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Springer-Verlag, Berlin, Heidelberg, 1997. MR 1479170 (98m:65007)
  • 27. V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, G-convergence of parabolic operators, Uspekhi Mat. Nauk 36: 1 (1981), 11-58. English Translation: Russ. Math. Surv. 36 (1981), 9-60. MR 0608940 (83a:35055)
  • 28. V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Averaging of parabolic operators, Trudy Mosk. Mat. O.-va 45 (1982), 182-236. English Translation: Trans. Moscow Math. Soc. 45 (1984), 189-241. MR 0704631 (85k:35024)
  • 29. V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, Heidelberg, 1994. MR 1318242 (96h:35003a) and 1329546 (96h:35003b)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N30, 35K05, 65N15

Retrieve articles in all journals with MSC (2000): 65N30, 35K05, 65N15

Additional Information

Pingbing Ming
Affiliation: LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS, Chinese Academy of Sciences, No. 55 Zhong-Guan-Cun East Road, Beijing, 100080, People’s Republic of China

Pingwen Zhang
Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

Keywords: Heterogeneous multiscale method, parabolic homogenization problems, finite element methods
Received by editor(s): June 3, 2003
Received by editor(s) in revised form: December 6, 2005
Published electronically: October 10, 2006
Additional Notes: The first author was partially supported by the National Natural Science Foundation of China under the grant 10571172 and also supported by the National Basic Research Program under the grant 2005CB321704.
The second author was partially supported by National Natural Science Foundation of China for Distinguished Young Scholars 10225103 and also supported by the National Basic Research Program under the grant 2005CB321704.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society