Numerical evaluation via singularity cancellation schemes of near-singular integrals involving the gradient of Helmholtz-type potentials

In this work we present a purely numerical procedure to evaluate strongly near-singular integrals involving the gradient of Helmholtz-type potentials for observation points at finite, arbitrarily small distances from the source domain. In the proposed approach the source domain is subdivided into a disc plus truncated subtriangles, and proper variable transformations are applied in each integration domain to exactly cancel the kernel singularity. A novel feature of the proposed angular transform is that required discrete values of the inverse transform, which is transcendental, are determined via a rootfinding procedure; the same idea can also be applied to other transforms that arise in singularity cancellation methods. The resulting integral may then evaluated via a low order Gauss- Legendre quadrature scheme.