In this paper we consider the Laplace-Dirichlet equation in a
polygonal domain perturbed at the scale
epsilon near one of
its vertices. We assume that this perturbation is self-similar, that
is, derives from the same pattern for all values of
epsilon.
We construct and validate asymptotic expansions of the solution in
powers of epsilon
via two different techniques, namely the
method of matched asymptotic expansions and the method of multiscale
expansions.
Then we show how the terms of each expansion can be split into a finite number of sub-terms in order to reconstruct the other expansion. Compared with the fairly general approach of Mazya, Nazarov and Plamenevskij [Birkhäuser, 2000] relying on multiscale expansions, the novelty of our paper is the rigorous validation of the method of matched asymptotic expansions, and the comparison of its result with that of the multiscale method. The consideration of a model problem allows to simplify the exposition of these rather complicated two techniques.
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Paper:
PDF file (364 k)
Preprint IRMAR 07-04 (2007). |
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Slides: PDF file (128 k)
Bad Herrenalb Workshop on Composites, 28 November 2006. |
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Note CRAS:
PDF file (148 k)
C. R. Acad. Sc. Paris Ser. I 343 (2006), pp 637-642. |
