Boundary integral method for the Dirichlet problem

Summary

The boundary integral method solves the Dirichlet problem for the Laplace equation with a double layer potential. The unknown harmonic function is written as that potential, with an unknown density on the boundary. The jump relations turn the boundary condition into a Fredholm integral equation of the second kind for the density. The Fredholm alternative then reduces existence of a solution to uniqueness. Its discretisation is the boundary element method of numerical analysis.