#### Vol. 2, No. 2, 2020

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

### Jeremy Hoskins and Manas Rachh

Vol. 2 (2020), No. 2, 447–476
##### Abstract

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.

##### Keywords
boundary integral equations, multiple junction interfaces, corners, singular solutions, potential theory
##### Mathematical Subject Classification 2010
Primary: 31A10, 35Q60, 45L05, 65E05, 65R20
##### Milestones
Received: 1 August 2019
Revised: 12 February 2020
Accepted: 16 March 2020
Published: 22 May 2020
##### Authors
 Jeremy Hoskins Applied Mathematics Program Yale University New Haven, CT United States Manas Rachh Center for Computational Mathematics Flatiron Institute New York, NY United States