  The present work deals with the resolution of the Poisson equation in a
bounded domain made of a thin and periodic layer of finite length placed into a
homogeneous medium. We provide and justify a high order asymptotic expansion
which takes into account the boundary layer effect occurring in the vicinity of
the periodic layer as well as the corner singularities appearing in the
neighborhood of the extremities of the layer. Our approach combines the method
of matched asymptotic expansions and the method of periodic surface
homogenization, and a complete justification is included in the paper or its
appendix.
