An integral equation method for solving Laplace’s equation with Robin boundary conditions

Boundary integral equations have been used to create effective methods for solving elliptic partial differential equations. Of primary importance is choosing the appropriate boundary representation for the solution such that the resulting integral equation is well-conditioned and solvable. Traditional boundary representations for Laplace’s equation use a double layer potential for Dirichlet problems and a single layer potential for Neumann problems since both lead to Fredholm integral equations of the second kind with continuous kernels. We investigate a representation that gives rise to Fredholm equations of the second kind for Robin boundary conditions. The equations have singular kernels for which we use specialized quadrature rules to construct numerical approximations. We also study the solution and conditioning of these methods with weak Robin conditions that approach Dirichlet ones in the limit and for domains that are multiply connected.