High-order Accurate Nyström Discretization of Integral Equations with Weakly Singular Kernels on Smooth Curves in the Plane

Boundary integral equations and Nyström discretization provide a powerful tool for the solution of Laplace and Helmholtz boundary value problems. However, often a weaklysingular kernel arises, in which case specialized quadratures that modify the matrix entries near the diagonal are needed to reach a high accuracy. We describe the construction of four different quadratures which handle logarithmically-singular kernels. Only smooth boundaries are considered, but some of the techniques extend straightforwardly to the case of corners. Three are modifications of the global periodic trapezoid rule, due to Kapur–Rokhlin, to Alpert, and to Kress. The fourth is a Kolm–Rokhlin modification to a quadrature based on Gauss-Legendre panels, which allows adaptivity. We compare in numerical experiments the convergence of the four schemes in various settings, including lowand high-frequency planar Helmholtz problems, and 3D axisymmetric Laplace problems. We also find striking differences in performance in an iterative setting. We summarize the relative advantages of the schemes.