Close evaluation of layer potentials in three dimensions

We present a simple and effective method for computing double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method is comprised of three steps: (i) rotate the spherical coordinate system so that the peak of the kernel in the nearly-singular integral is aligned with its north pole; (ii) integrate first with respect to azimuth using the Periodic Trapezoid Rule, which is a natural averaging operation yielding a relatively smooth function of the polar angle; and (iii) integrate with respect to the polar angle (rather than the cosine of the polar angle) using an open, Gauss-Legendre quadrature rule mapped to [0, π]. We find that the asymptotic behavior of the error for this method in the limit of small evaluation distance from the boundary is quadratic for the modified double-layer potential after applying a subtraction method to it, and linear for the single-layer potential. Additionally, we use this method for computing the matrix entries of the Galerkin method to solve the underlying boundary integral equations. We illustrate the accuracy and utility of this method through several numerical examples