A regularized boundary integral method in potential theory

A global method for solving the boundary integral equation in potential theory is presented. With subtracting and adding back treatment, several auxiliary functions are employed to reduce the singularities of boundary integrals. After that all the integrals can be directly integrated by arbitrary numerical quadrature without any geometric approximation. The advantage of this method is that high accuracy of numerical quadrature can be maintained with minimal computational nodes in boundary integral, since collocation points can be arranged exactly as quadrature points. The proposed auxiliary functions can also smooth the peaks of nearly singular integrals if they exist. In such a case, little extra work is needed to separate singular and nearly singular integrals into different elements. After separation, both kinds of integrals can be partially transformed into line integrals along their boundaries for the three-dimensional case or several simple formulas for the two-dimensional case.