Efficient quadrature rules based on spline quasi--interpolation for application to IGA-BEMs

We propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals, based on the spline quasi-interpolation scheme introduced in Mazzia and Sestini (2009). These integrals in particular occur in the entries of the stiffness matrix coming from Isogeometric Boundary Element Methods (IgA-BEMs). The presented formulas are efficient, since they combine the locality of any spline quasi-interpolation scheme with the capability to compute the modified moments for B-splines, i.e. the weakly or strongly singular integrals of such functions. No global linear system has to be solved to determine the quadrature weights, but just local systems, whose size linearly depends on the adopted spline degree. The rules are preliminarily tested in their basic formulation, i.e. when the integrand is defined as the product of a singular kernel and a continuous function g. Then, such basic formulation is compared with a new one, specific for the approximation of the singular integrals appearing in the IgA-BEM context, where a B-spline factor is explicitly included in g. Such a variant requires the usage of the recursive spline product formula given in Mørken (1991), and it is useful when the ratio between g and its B-spline factor is smooth enough.