Interior Error Estimate for Periodic Homogenization

Abstract

Abstract.In a previous article about the homogenization of the classical problem of diffusion in a bounded domain with sufficiently smooth boundary we proved that the error is of order ε 1/2 . Now, for an open set Ω with sufficiently smooth boundary (C

Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

“…Applying estimates in Lemmata 2.3, 2.5 and Convergence Theorem [7,8], see Theorem 5.3 in Appendix, implies the convergences for u ε , v ε , w ε in (12). The strong convergence of r ε is achieved by showing that T…”

“…The main ideas of the methodology were presented in [12,13] and applied to linear elliptic equations with oscillating coefficients, posed in a fixed domain. Our approach strongly relies on these results.…”

Norm-Resolvent Convergence of One-Dimensional High-Contrast Periodic Problems to a Kronig–Penney Dipole-Type Model

“…The formula (30) suggests that after the unitary rescaling ε , the differential expression that defines the operator loses its dependence on the parameter ε on the soft component. This becomes obvious after the substitution τ = εt in (30).…”

“…The formula (30) suggests that after the unitary rescaling ε , the differential expression that defines the operator loses its dependence on the parameter ε on the soft component. This becomes obvious after the substitution τ = εt in (30). Henceforth, we use τ and εt interchangeably: τ in the objects pertaining to the soft component, and εt in those pertaining to the stiff component, as in the latter case one cannot drop the explicit dependence on ε.…”

“…In conclusion, we mention some papers that considered the norm-resolvent convergence for operators with periodic rapidly oscillating coefficients: in [10,30,35,47,56,58] the authors established sharp estimates for the rates of convergence in the sense of various operator norms. Norm-resolvent convergence was established also for certain perturbations in the boundary homogenisation: in [11,12], where problems with frequent alternation (periodic and non-periodic) of boundary conditions were treated, in [15], [46,Ch.…”