Numerical Quadratures for Singular and Hypersingular Integrals in Boundary Element Methods

A method is developed for the computation of the weights and nodes of a numerical quadrature which integrates functions containing singularities up to order $1/x^{2}$, without the requirement to know the coefficients of the singularities exactly. The work is motivated by the need to evaluate such integrals on boundary elements in potential problems and is a simplification of a previously published method, but with the advantage of handling singularities at the endpoints of the integral. The numerical performance of the method is demonstrated by application to an integral containing logarithmic, first, and second order singularities, characteristic of the problems encountered in integrating a Green's function in boundary element problems. It is found that the quadrature is accurate to 11--12 decimal places when computed in double precision.