On a quadrature formula for the simple layer potential in a 3D case

A quadrature formula for the simple layer potential with smooth density specified on a closed or open surface is derived for a wide class of smooth surfaces including sphere and ellipsoid. The formula uniformly approximates the potential near the surface and saves its continuity as the observation point tends to the surface from the interior of the domain, that is confirmed by numerical tests. The proposed quadrature formula is much more accurate in calculating the potential near the surface than the standard quadrature formulas, that is confirmed by numerical tests also. In addition, a quadrature formula for direct value of the simple layer potential on a surface is derived. Numerical tests for this formula are also given and they confirm its efficiency and accuracy.