  In this paper we characterize the behavior of solutions to systems of
boundary integral equations associated with Laplace transmission problems in
composite media consisting of regions with polygonal boundaries. In particular
we consider triple junctions, i.e. points at which three distinct media meet.
We show that, under suitable conditions, solutions to the boundary integral
equations in the vicinity of a triple junction are well-approximated by linear
combinations of functions of the form $t^\beta,$ where $t$ is the distance of
the point from the junction and the powers $\beta$ depend only on the material
properties of the media and the angles at which their boundaries meet.
Moreover, we use this analysis to design efficient discretizations of boundary
integral equations for Laplace transmission problems in regions with triple
junctions and demonstrate the accuracy and efficiency of this algorithm with a
number of examples.
