High order convergence for collocation of second kind boundary integral equations on polygons

We propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon. They are based on the bijectivity of the double layer potential between spaces of Sobolev type with arbitrary high regularity and involving the singular functions generated by the corners, If splines of order 2m - 1 are used, we get quasi-optimal estimates in H-m-norm and optimal order convergence for the H-k-norm if 0 less than or equal to k less than or equal to m. Numerical experiments are presented.