A Strange Term in the Homogenization of Parabolic Equations with Two Spatial and Two Temporal Scales

L Flodén,A Holmbom,M Olsson Lindberg

doi:10.1155/2012/643458

Abstract

We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficienta(x/ε,t/ε2)in the elliptic part and spatial oscillations in the coefficientρ(x/ε)that is multiplied with the time derivative∂tuε. We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation inρ(x/ε)and the temporal oscillation ina(x/ε,t/ε2)and disappears if either of these oscillations is removed.

Highlights

We study the homogenization of ρ x ε ∂tuε x, t − ∇ · a x ε, t ε2∇uε x, t f x, t in Ω × 0, T, uε x, 0 g x in Ω, 1.1 uε x, t 0 on ∂Ω × 0, T, which contains oscillations in both space and time in the coefficient a x/ε, t/ε2 in the elliptic part and spatial oscillations in the coefficient ρ x/ε that is multiplied with the time derivative ∂tuε
We study the homogenization of a parabolic equation with oscillations in both space and time in the coefficient a x/ε, t/ε2 in the elliptic part and spatial oscillations in the coefficient ρ x/ε that is multiplied with the time derivative ∂tuε
We obtain a strange term in the local problem. This phenomenon appears as a consequence of the combination of the spatial oscillation in ρ x/ε and the temporal oscillation in a x/ε, t/ε2 and disappears if either of these oscillations is removed

Summary

Introduction

∇uε x, t f x, t in Ω × 0, T , uε x, 0 g x in Ω, 1.1 uε x, t 0 on ∂Ω × 0, T , which contains oscillations in both space and time in the coefficient a x/ε, t/ε2 in the elliptic part and spatial oscillations in the coefficient ρ x/ε that is multiplied with the time derivative ∂tuε. To deal with the oscillations of ρ x/ε , we need to make a special choice of test functions for our approach to apply, which is the reason why an additional term is obtained in the local problem. This phenomenon appears as a consequence of the combination. A related problem is studied by Nandakumaran and Rajesh in 1 , with the temporal oscillations of the same frequency as the spatial ones and the resonance phenomenon in the local problem that we obtain for 1.1 does not appear; see Remarks 3.3 and 3.4. Simpler linear problems without temporal oscillations are found in, for example, 2, 3

Two-Scale Convergence

Homogenization

To find a local problem we choose vx εv1 x v x ε