Bloch approximation in homogenization on bounded domains

Abstract

The classical problem of homogenization deals with elliptic operators in periodically oscillating media of small period ε>0 and the asymptotic behavior of solution u ε of boundary value problems associated with such operators. In a previous work [Conca et al., SIAM J. Math. Anal. 33 (2002), 1166–1198], the authors introduced what is called Bloch approximation which provided energy norm approximation for the solution in $\mathbb{R}^{N}$ . This paper continues with the above development and proposes a modified Bloch approximation. This function takes into account boundary effects. Its connection with the first order classical correctors is also established with the corresponding error estimate. All the proofs are worked out entirely in the Fourier–Bloch space.