In this paper we consider the Laplace-Dirichlet equation in a polygonal domain perturbed at the small scale $\varepsilon$ near a vertex. We assume that this perturbation is self-similar, that is, derives from the same pattern for all relevant values of $\varepsilon$. We construct and validate asymptotic expansions of the solution in powers of $\varepsilon$ via two different techniques, namely the method of multiscale expansions and the method of matched asymptotic 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 Maz'ya, Nazarov and Plamenevskii relying on multiscale expansions, the novelty of our paper is the rigorous validation of the method of matched asymptotic expansions, and its comparison with the multiscale method. The consideration of a model problem allows to simplify the exposition of these rather complicated two techniques.
In Around the Research of Vladimir Maz'ya II,
Partial Differential Equations.
International Mathematical Series (Springer) Vol. 12, pp 95-134 (2010).
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Slides: PDF file (128 k)
Bad Herrenalb Workshop on Composites, 28 November 2006.
PDF file (148 k)
C. R. Acad. Sc. Paris Ser. I 343 (2006), pp 637-642.