A novel high-order Nystrom scheme for 3D boundary integral equations

The traditional Nystrom method is designed to treat regular integral kernels, whereas our boundary integral equations (BIE) present singular or hyper-singular kernels associated with the Green's functions. This problem has been solved by using the local corrections with a high-order accuracy for the singular or near-interaction terms. Although these local corrections are easy and efficient for simple cases, they may not be convenient to implement for 3D cases with curvilinear geometries. We develop a novel high-order Nystrom scheme based on the Lagrange interpolation for the unknown function. The scheme is easy to implement and the resulting error is controllable.