Selfsimilar perturbation near a corner: Matching and Multiscale expansions

Sébastien Tordeux (1) , Grégory Vial (2) , Monique Dauge (2)

1 MIP - Mathématiques pour l'Industrie et la Physique
2 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper we consider the Laplace-Dirichlet equation in a polygonal domain perturbed at the scale $\varepsilon$ near one of its vertices. We assume that this perturbation is self-similar, that is, derives from the same pattern for all values of $\varepsilon$. We construct and validate asymptotic expansions of the solution in powers of $\varepsilon$ 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 & Plamenevskij 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.

Domaines: Equations aux dérivées partielles [math.AP]

Sébastien Tordeux, Grégory Vial, Monique Dauge. Selfsimilar perturbation near a corner: Matching and Multiscale expansions. 2007. ⟨hal-00124936v1⟩