Two general methods for the numerical approximation of multidimensional Cauchy principal value integrals

The numerical approximation of integrals containing strongly singular integrals, in particular Cauchy principal value integrals, is a major issue connected, e.g., to the boundary integral approach for many types of partial differential equations. Whereas the one-dimensional problem has been addressed very intensively in recent years, much less attention has been paid to multidimensional problems. In the present paper, we investigate two possible approaches to this problem, corresponding to generalizations of two approaches known in the 1-D case. In principle, both methods can be applied to integration domains of arbitrary shape, although we find that certain combinations of algorithms and domains are more useful than others. In particular, we discuss error estimates and show that the methods are highly competitive. Moreover, in contrast to most of the previously discussed methods, the approaches are very efficient when integrals have to be calculated for various locations of the singularity