Quadrature methods for Symm's integral equation on polygons

In a recent paper the authors obtained stability and convergence results for spline collocation methods of arbitrarily high order applied to the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a polygonal domain. This paper obtains the same convergence results for the case when the collocation integrals are approximated using an appropriate quadrature rule. The analysis depends on the calculus of non-standard Mellin pseudodifferential operators and on some new estimates for the kernels of these operators. Calculations using piecewise constant and piecewise linear splines are reported. These are consistent with the theoretical results and suggest some further conjectures about the performance of this method.