2012/12/05 by T. A. Suslina, Suslina, Tatiana
Computer Science · Engineering · #35B27 (Primary) #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1212.1148
openalex publication_date 2012/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathcal O ⊂ \mathbb Rd be a bounded domain with the boundary of class C1,1. In L2(\mathcal O;\mathbb Cn), a matrix elliptic second order differential operator \mathcal AN,ε with the Neumann boundary condition is considered. Here ε>0 is a small parameter, the coefficients of \mathcal AN,ε are periodic and depend on \mathbf x /ε. There are no regularity assumptions on the coefficients. It is shown that the resolvent (\mathcal AN,ε+λI)-1 converges in the L2(\mathcal O;\mathbb Cn)-operator norm to the resolvent of the effective operator \mathcal AN0 with constant coefficients, as ε → 0. A sharp order error estimate |(\mathcal AN,ε+λI)-1 - (\mathcal AN0 +λI)-1|L2→ L2 ≤ Cε is obtained. Approximation for the resolvent (\mathcal AN,ε+λI)-1 in the norm of operators acting from L2(\mathcal O;\mathbb Cn) to the Sobolev space H1(\mathcal O;\mathbb Cn) with an error O(√(ε)) is found. Approximation is given by the sum of the operator (\mathcal A0N +λI)-1 and the first order corrector. In a strictly interior subdomain \mathcal O' a similar approximation with an error O(ε) is obtained.