Methods for computing singular and nearly singular integrals

Many scientific problems are formulated in terms of singular integrals. We describe a simple method for computing such integrals. Our approach is to replace a singularity, or near singularity, with a regularized version, compute a sum in a standard way, and then add correction terms, which are found by asymptotic analysis near the singularity. We have used this approach for a single-layer potential on a doubly periodic surface, evaluated at a grid point on the surface. The quadrature rule so developed was used to design a convergent boundary integral method for three-dimensional water waves. In related work we have developed a method for computing a double-layer potential on a curve, evaluated at a point near the curve. Thus values of harmonic functions, with prescribed boundary conditions on a curve, can be calculated at grid points inside or outside, with only slightly extra effort for those points near the boundary. This procedure may be useful for computing fluid flow with moving boundaries. This arti...