Analysis of the heterogeneous multiscale method for parabolic homogenization problems
HTML articles powered by AMS MathViewer
- by Pingbing Ming and Pingwen Zhang PDF
- Math. Comp. 76 (2007), 153-177 Request permission
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
- Assyr Abdulle and Weinan E, Finite difference heterogeneous multi-scale method for homogenization problems, J. Comput. Phys. 191 (2003), no. 1, 18–39. MR 2008485, DOI 10.1016/S0021-9991(03)00303-6
- G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), no. 4, 691–727 (English, with English and French summaries). MR 1178564, DOI 10.4153/CJM-1992-042-x
- N. André and M. Chipot, Uniqueness and nonuniqueness for the approximation of quasilinear elliptic equations, SIAM J. Numer. Anal. 33 (1996), no. 5, 1981–1994. MR 1411859, DOI 10.1137/S0036142994267400
- Michel Artola and Georges Duvaut, Un résultat d’homogénéisation pour une classe de problèmes de diffusion non linéaires stationnaires, Ann. Fac. Sci. Toulouse Math. (5) 4 (1982), no. 1, 1–28 (French, with English summary). MR 673637
- N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788, DOI 10.1007/978-94-009-2247-1
- Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
- Lucio Boccardo and François Murat, Remarques sur l’homogénéisation de certains problèmes quasi-linéaires, Portugal. Math. 41 (1982), no. 1-4, 535–562 (1984) (French, with English summary). MR 766874
- S. Brahim-Otsmane, G. A. Francfort, and F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl. (9) 71 (1992), no. 3, 197–231. MR 1172450
- Shanqin Chen, Weinan E, and Chi-Wang Shu, The heterogeneous multiscale method based on the discontinuous Galerkin method for hyperbolic and parabolic problems, Multiscale Model. Simul. 3 (2005), no. 4, 871–894. MR 2164241, DOI 10.1137/040612622
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Ferruccio Colombini and Sergio Spagnolo, Sur la convergence de solutions d’équations paraboliques, J. Math. Pures Appl. (9) 56 (1977), no. 3, 263–305 (French, with English summary). MR 603300
- Andrea Dall’Aglio and François Murat, A corrector result for $H$-converging parabolic problems with time-dependent coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 329–373 (1998). Dedicated to Ennio De Giorgi. MR 1655521
- Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI 10.1137/0707048
- Weinan E, Analysis of the heterogeneous multiscale method for ordinary differential equations, Commun. Math. Sci. 1 (2003), no. 3, 423–436. MR 2069938
- Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. MR 1979846
- Weinan E and Bjorn Engquist, Multiscale modeling and computation, Notices Amer. Math. Soc. 50 (2003), no. 9, 1062–1070. MR 2002752
- Weinan E, Pingbing Ming, and Pingwen Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156. MR 2114818, DOI 10.1090/S0894-0347-04-00469-2
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- J. García-Azorero, C. E. Gutiérrez, and I. Peral, Homogenization of quasilinear parabolic equations in periodic media, Comm. Partial Differential Equations 28 (2003), no. 11-12, 1887–1910. MR 2015406, DOI 10.1081/PDE-120025489
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
- Sergio Spagnolo, Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 657–699 (Italian). MR 225015
- S. Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. , Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 571-597; errata, ibid. (3) 22 (1968), 673 (Italian). MR 0240443
- Sergio Spagnolo, Convergence of parabolic equations, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 2, 547–568 (English, with Italian summary). MR 0460889
- François Murat and Luc Tartar, $H$-convergence, Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, Birkhäuser Boston, Boston, MA, 1997, pp. 21–43. MR 1493039
- Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170, DOI 10.1007/978-3-662-03359-3
- V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, $G$-convergence of parabolic operators, Uspekhi Mat. Nauk 36 (1981), no. 1(217), 11–58, 248 (Russian). MR 608940
- V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Averaging of parabolic operators, Trudy Moskov. Mat. Obshch. 45 (1982), 182–236 (Russian). MR 704631
- V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242
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
- Email: mpb@lsec.cc.ac.cn
- Pingwen Zhang
- Affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
- Email: pzhang@pku.edu.cn
- 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. - © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 153-177
- MSC (2000): Primary 65N30, 35K05, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-06-01909-0
- MathSciNet review: 2261016