On the rate of convergence of solutions in domain with periodic multilevel oscillating boundary

Abstract

In this paper we deal with the homogenization problem for the Poisson equation in a singularly perturbed domain\ud with multilevel periodically oscillating boundary. This domain consists of the body, a large number of thin cylinders\ud joining to the body through the thin transmission zone with rapidly oscillating boundary. Inhomogeneous Fourier\ud boundary conditions with perturbed coefficients are set on the boundaries of the thin cylinders and on the boundary\ud of the transmission zone. We prove the homogenization theorems and derive the estimates for the convergence of the\ud solutions

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

“…We refer the reader to [7] for related discussion of continuum mechanical descriptions of balance laws in continua with microstructure and to [28] for a few workedout averaging examples away from the periodic setting. At the mathematical level, we succeed in combining successfully the philosophy of getting correctors as explained in the analysis by Chechkin and Piatnitski [10] with the intimate two-scale structure of our system; see also [8,9] for related settings where similar averaging strategies are used. For methodological hints on how to get convergence rates for homogenization-like limits, we refer the reader to the analysis shown in [16] for the case of a reaction-diffusion phase field-like system posed in fixed domains.…”

Convergence Rates in Homogenization of the Mixed Boundary Value Problems

“…In homogenization problems for the Poisson equations in a domain with oscillating boundary, the 1 convergence rate have been studied in [13][14][15], while work [16] deals with the multilevel oscillation of the boundary with different conflicting boundary conditions. See also [17][18][19] for more results on the asymptotic behavior of eigenvalues for the boundary value problems in domains with oscillating boundaries or interfaces.…”