Modified Representations for the Close Evaluation Problem

When using boundary integral equation methods, we represent solutions of a linear partial differential as layer potentials. It is well-known that the approximation potentials quadrature rules suffer from poor resolution when evaluated closed to (but not on) boundary. To address this challenge, provide modified representations problem’s solution. Similar Gauss’s law used modify Laplace’s double-layer potential, use single-layer potential and Helmholtz avoid close evaluation problem. Some techniques have been developed in context representation formula or interpolation techniques. We alternative directly (or only one density at stake). Several numerical examples illustrate efficiency technique two three dimensions.