Advantages of the Normal Derivative Method to Set Up the Integral Kernels for a Unit Disc

Abstract

We implement the boundary element method for the Helmholtz equation for a two-dimensional unit disc with Dirichlet boundary conditions. This yields boundary integral equations of Fredholm kind. Here we discuss the advantages of the normal derivative method which leads to a second-kind Fredholm integral equation instead of dealing with the resulting first-kind Fredholm integral equation. This is shown by comparing the accuracy of the boundary functions which are computed using both types of integral equations. We point out that it may be advantageous for smooth boundary to use the normal derivative method to set up the boundary integral kernels which gets rid of the weakly-singular integrals..