# On the solution of Laplace's equation in the vicinity of triple-junctions

## TL;DR

This paper analyzes the behavior of solutions to Laplace boundary integral equations near triple junctions in composite media, providing a theoretical characterization and developing efficient numerical discretization methods.

## Contribution

It characterizes solution behavior near triple junctions and introduces an efficient discretization approach for boundary integral equations in such regions.

## Key findings

- Solutions near triple junctions are well-approximated by functions of the form t^β.
- The powers β depend only on material properties and boundary angles.
- The proposed discretization method demonstrates high accuracy and efficiency.

## Abstract

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.