Abstract
Reiterated homogenization is studied for divergence structure parabolic problems of the form ∂ u ɛ/∂t−div (a(x,x/ɛ,x/ɛ2,t,t/ɛ k)∇u ɛ)=f. It is shown that under standard assumptions on the function a(x, y 1,y 2,t,τ) the sequence {u ɛ} of solutions converges weakly in L 2 (0,T; H 10 (Ω)) to the solution u of the homogenized problem ∂u/∂t− div(b(x,t)∇u)=f.
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References
G. Allaire, M. Briane: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb. 126 (1996), 297–342.
M Avellaneda: Iterated homogenization, differential effective medium theory and applications. Commun. Pure Appl. Math. 40 (1987), 527–554.
A. Bensoussan, J.-L. Lions, and G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam-New York-Oxford, 1978.
D Cioranescu, P. Donato: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications. Oxford Univ. Press, New York, 1999.
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.
A Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results. Appl. Math. 42 (1997), 321–343.
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.
N. Svanstedt, N. Wellander: A note on two-scale convergence of differential operators. Submitted.
R. Temam: Navier-Stokes equations. Theory and Numerical Analysis. North-Holland, Amsterdam-New York-Oxford, 1977.
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Holmbom, A., Svanstedt, N. & Wellander, N. Multiscale convergence and reiterated homogenization of parabolic problems. Appl Math 50, 131–151 (2005). https://doi.org/10.1007/s10492-005-0009-z
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DOI: https://doi.org/10.1007/s10492-005-0009-z