Fibre homogenisation

Abstract

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by 'fibrewise' homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods. Abstract fibre homogenisationLet Θ be a non-empty set. For a given family of Hilbert spaces (H θ ) θ∈Θ , ε ∈ (0, ∞), M(θ) ∈ L(H θ ) with M ∞ := sup θ∈Θ M(θ) < ∞, and A(θ) : dom(A(θ)) ⊆ H θ → H θ densely defined and closed, we consider the operator family B ε (θ) := M(θ) + 1 ε A(θ).

Resolvent estimates and numerical implementation for the homogenisation of one‐dimensional periodic mixed type problems

“…In [], only a qualitative convergence statement was derived. The techniques developed in [], however, suggest that the rate of convergence can be quantified. It is one main result of the present exposition – based on the rationale outlined in [] – that a quantified convergence rate for problems of the type discussed in [] can be derived.…”

Nonlocal H-convergence

“…Next, from (11), we obtain π 0 w = a 00 π 0 v + a 01 π 1 v. Hence, a 11 − a 10 a −1 00 a 01 π 1 v = π 1 w − a 10 a −1 00 a 00 π 0 v + a 01 π 1 v = π 1 w − a 10 π 0 v − a 10 a −1 00 a 01 π 1 v.…”

“…where the last equation follows from Lemma 4.8(b). With this, we infer a n A 0 u n = a n UU * A 0 u n = U a n,00 a n,01 a n,10 a n, 11 A 0 u n 0 = U a n,00 a n,01 a n,10 a n,11…”

“…So, let n ∈ N and let f and g be as in the definition of nonlocal H-convergence and let u n and v n be the corresponding solutions. Then we have 11 − a n,10 a −1 n,00 a n,01 ((A * 1 ) ⋄ ) −1 g…”

Homogenisation and the weak operator topology

“…Furthermore, applications to fractional elasticity have been found in [33]; applications to highly oscillatory mixed type problems have been discussed in [37,9,8]. Quantitative results (optimal in order, operator-norm estimates) in this line of problems using compactness of the spatial derivative operator or a spectral gap type condition can be found in [5] (in arbitrary spatial dimensions) and in [2] (in a one-dimensional dynamic setting). Note that for quantitative estimates to be shown, in the mentioned results the periodicity of the coefficients is essential.…”