A higher-order singularity subtraction technique for the discretization of singular integral operators on curved surfaces

A major challenge facing the community working on integral equation based solvers for boundary value problems is the construction of efficient discretizations of integral operators with singular kernels on curved surfaces. Classic approaches such as singularity subtraction, special purpose quadrature, singularity cancellation, kernel regularization, and various adaptive strategies may work well in many situations but have not yet fully, in three dimensions, succeeded in unleashing the computational power of integral equation methods needed for excellence in real-world physics applications [5, Section 1]. Recently, two new promising methods have been launched: the quadrature by extension (QBX) method which exploits that fields induced by integral operators are often smooth close to the boundaries where their sources are located [5] and a method relying on a combination of adaptivity, local invertible affine mappings with certain orthogonality properties, and the use of precomputed tables of quadrature rules [1]. It seems to be an open question what method, or combination of techniques, is best. This note is about promoting a classic technique for the discretization of singular integral operators on curved surfaces, namely singularity subtraction. The idea is to use analytical evaluation to a maximum degree and split singular (and nearly singular) operators into two parts each – one illbehaved part whose action can be evaluated using high-order analytic product integration, and another more regular part for which purely numerical integration is used, compare [2]. Based on this idea we present and implement a simple Nyström scheme for Laplace’s equation on tori. Surprisingly accurate results are produced.