Asymptotic expansion of the solution of an interface problem in a polygonal domain with thin layer

Gabriel Caloz, Martin Costabel, Monique Dauge, Gregory Vial

We consider the solution of an interface problem posed in a bounded domain coated with a layer of thickness epsilon and with external boundary conditions of Dirichlet or Neumann type. Our aim is to build a multi-scale expansion as epsilon --> 0 for that solution.

After presenting a complete multi-scale expansion in a smooth coated domain, we focus on the case of a corner domain. Singularities appear, obstructing the construction of the expansion terms in the same way as in the smooth case. In order to take these singularities into account, we construct profiles in an infinite coated sectorial domain.

Combining together expansions in the smooth case with splittings in regular and singular parts involving the above mentioned profiles, we construct two families of multi-scale expansions for the solution in the coated domain with corner. We prove optimal estimates for the remainders of the multi-scale expansions.

This paper is based on techniques similar to those of CoDaSimix.html, although much more involved, since the present situation has a higher level of complexity.

Version of 19 December 2005: PDF file (544 k)
Revised version with hyperlinks.
Asymptotic Analysis Vol. 50, n° 1/2, (2006), pp 121-173.