Improved quadrature formulas for the direct value of the normal derivative of a single-layer potential

A single-layer potential for the Helmholtz equation in the three-dimensional case and a single-layer potential for the Laplace equation are considered. A quadrature rule is derived for the direct value of the normal derivative of the single-layer potential with a continuous density given on a closed or open surface. The quadrature rule provides a much higher accuracy than previously available formulas, which is confirmed by numerical tests. The quadrature rule can be used for the numerical solution of boundary value problems for Laplace and Helmholtz equations by applying the boundary integral equation method.