An optimal order collocation method for first kind boundary integralequations on polygons

This paper discusses the convergence of the collocation method usingsplinesof any order (Formula presented.) for first kind integral equations withlogarithmic kernelson closed polygonal boundaries in (Formula presented.). Beforediscretization theequation is transformed to an equivalent equation over(Formula presented.) using anonlinear parametrization of the polygon which varies more slowly thanarc--length near each corner. This has the effect of producing a transformedequation with a solution which is smooth on (Formula presented.). Thislatterintegral equation is shown to be well--posed in appropriate Sobolev spaces.The structure of the integral operator is described in detail, and can bewritten in terms of certain non--standard Mellin convolution operators.Using thisinformation we are able to show that the collocation method using splines oforder (Formula presented.) (degree (Formula presented.)) converges with optimal order (Formula presented.).(Thecollocation points are the mid--points of subintervals when (Formula presented.) isoddand the break--points when (Formula presented.) is even, and stability is shownunder theassumption that the method may be modified slightly.) Using the numericalsolutions to the transformed equation we construct numerical solutions of theoriginal equation which converge optimally in a certain weighted norm.Finally the method is shown to produce superconvergent approximations tointerior potentials such as those used to solve harmonic boundary valueproblems by the boundary integral method. The convergence results areillustrated with some numerical examples.