MATHEMATICS OF COMPUTATION Volume 76, Number 257, January 2007, Pages 153–177 S 0025-5718(06)01909-0 Article electronically published on October 10, 2006 ANALYSIS OF THE HETEROGENEOUS MULTISCALE METHOD FOR PARABOLIC HOMOGENIZATION PROBLEMS PINGBING MING AND PINGWEN ZHANG Abstract. The heterogeneous multiscale method (HMM) is applied to var- ious 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. 1. Introduction and main results 1.1. Generality. Consider the following parabolic problem: (1.1) ⎧ ⎪ ⎨ ⎪ ⎩ ∂ t u ε −∇· (a ε ∇u ε )= f in D × (0,T ) =: Q, u ε =0 on ∂D × (0,T ), u ε | t=0 = u 0 . Here ε is a small parameter that signifies the multiscale nature of the problem. We let D be a bounded domain in R d and T a positive real number. A problem of this type is interesting because of its simplicity and its relevance to several important practical problems, such as the flow in porous media and the mechanical properties of composite materials. In contrast to the elliptic problems there may be oscillations in the temporal direction besides the oscillation in the spatial direction. On the analytic side, the following fact is known about (1.1). In the sense of parabolic H-convergence (see [25], [8], [12]), introduced with minor modification by Spagnolo and Colombini under the name of G-convergence or PG-convergence (see [11], [22], [23], [24]), for every f ∈ L 2 (0,T ; H −1 (D)) and u 0 ∈ L 2 (D), the sequence {u ε } the solutions of (1.1) satisfies u ε U weakly in L 2 (0,T ; H 1 0 (D)), a ε ∇u ε  A∇U weakly in L 2 (Q; R d ), Received by the editor June 3, 2003 and, in revised form, December 6, 2005. 2000 Mathematics Subject Classification. Primary 65N30, 35K05, 65N15. Key words and phrases. Heterogeneous multiscale method, parabolic homogenization prob- lems, finite element methods. 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. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 153 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use