SINGULAR AND NEAR-SINGULAR INTEGRALS IN HIGH-PRECISION DERIVATIVE COMPUTATION

Accurate numerical differentiation of approximate data by methods base d on Green's second identity often involves singular or nearly singula r integrals over domains or their boundaries. This paper applies the f inite part integration concept to evaluate such integrals and to gener ate suitable quadrature formulae. The weak singularity involved in fir st derivatives is removable; the strong singularities encountered in c omputing higher derivatives can be reduced. To find derivatives on or near the edge of the integration region, special treatment of boundary integrals is required. Values of normal derivative at points on the e dge are obtainable by the method described. Example results are given for derivatives of analytically known functions, as well as results fr om finite element analysis.