A new first kind boundary integral formulation for the Dirichlet-to-Neumann map in 2D

In this paper, we analyze the Dirichlet-to-Neumann (DtN) operator in the periodic case as a pseudodifferential operator represented through boundary integrals. We begin with some analytical results concerning the structure of the operator. In particular we exploit the freedom available in the choice of the kernel for the boundary integral representation to introduce a new logarithmic kernel for the fundamental solution of the Laplacian on a cylinder. We then use it to develop a superalgebraically convergent numerical method to compute DtN which proves very stable even for nonsmooth and large variation curves. An important step in the proposed procedure is the inversion of an integral equation of first kind. To deal with it, we introduce an efficient FFT-based preconditioner which performs well in combination with Nystrom’s method and a decomposition of the operator based on “flat geometry subtraction”.