Optimal order collocation for the mixed boundary value problem on polygons

In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not $1$. This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order $2m-1$ are used, we get quasi-optimal estimates in $H^m$-norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.