Holmbom, Anders; Svanstedt, Nils; Wellander, Niklas Multiscale convergence and reiterated homogenization of parabolic problems. (English) Zbl 1099.35011 Appl. Math., Praha 50, No. 2, 131-151 (2005). Summary: Reiterated homogenization is studied for divergence structure parabolic problems of the form \(\partial u_{\varepsilon }/\partial t - \text{div}\bigl (a\bigl (x, x/\varepsilon ,x/\varepsilon ^2, t,t/\varepsilon ^{k}\bigr )\nabla u_{\varepsilon }\bigr )=f\). It is shown that under standard assumptions on the function \(a(x,y_1,y_2,t,\tau )\) the sequence \(\{u_\varepsilon \}\) of solutions converges weakly in \(L^2(0,T;H^1_0(\Omega ))\) to the solution \(u\) of the homogenized problem \(\partial u/\partial t - \text{div} (b(x,t)\nabla u)=f\). Cited in 27 Documents MSC: 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:reiterated homogenization; multiscale convergence; parabolic equation × Cite Format Result Cite Review PDF Full Text: DOI EuDML Link Geodesic References: [1] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb. 126 (1996), 297–342. · Zbl 0866.35017 [2] M Avellaneda: Iterated homogenization, differential effective medium theory and applications. Commun. Pure Appl. Math. 40 (1987), 527–554. · Zbl 0629.73010 · doi:10.1002/cpa.3160400502 [3] A. Bensoussan, J.-L. Lions, and G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam-New York-Oxford, 1978. · Zbl 0404.35001 [4] D Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications. Oxford Univ. Press, New York, 1999. · Zbl 0939.35001 [5] A. Dall’Aglio, F. Murat: A corrector result for H-converging parabolic problems with time-dependent coe-cients. Dedicated to Ennio De Giorgi. Ann. Sc. Norm. Super. Pisa Cl. Sci. IV 25 (1997), 329–373. [6] A Holmbom: Homogenization of parabolic equations–an alternative approach and some corrector-type results. Appl. Math. 42 (1997), 321–343. · Zbl 0898.35008 · doi:10.1023/A:1023049608047 [7] J.-L. Lions, D. Lukkassen, L. E. Persson, and P. Wall: Reiterated homogenization of nonlinear monotone operators. Chin. Ann. Math. Ser. B 22 (2001), 1–12. · Zbl 0979.35047 · doi:10.1142/S0252959901000024 [8] N. Svanstedt, N. Wellander: A note on two-scale convergence of differential operators. Submitted. [9] R. Temam: Navier-Stokes equations. Theory and Numerical Analysis. North-Holland, Amsterdam-New York-Oxford, 1977. · Zbl 0383.35057 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.