Uniform asymptotics of Green's kernels in perforated domains and meso-scale approximations

Abstract

The article is a review of the authors' results on asymptotic approximations of Green's kernels for elliptic boundary value problems in perforated domains. A new feature is the uniformity of the asymptotics with respect to the independent variables. Formal asymptotic approximations are supplied with estimates of the remainder terms. For the case when the number of perforations or inclusions becomes large, a novel method of meso-scale asymptotic approximations is discussed, and uniform asymptotic approximations of Green's kernels as well as solutions of boundary value problems in multiply perforated domains are presented. Such approximations do not require periodicity or other typical constraints attributed to homogenization approximations.

Keywords:

• uniform asymptotic approximations
• Green's functions
• singularly perturbed domains

AMS Subject Classifications::

• 35Q70
• 34E05
• 34E10

Acknowledgement

The support of the UK Engineering and Physical Sciences Research Council via the grant EP/F005563/1 is gratefully acknowledged.

Notes

1. The computational examples in Figures 3 and 4 have been produced by Dr M. Nieves.