The numerical solution of Symmʹs equation on smooth open arcs by spline Galerkin methods

We consider the classical Fredholm linear integral equation of the first kind with logarithmic kernel on a smooth Jordan open arc. Applying the well-known cosine change of variable, the arc is reparametrized and the problem is transformed into a new integral equation. We investigate the existence of an asymptotic expansion for the error of the Galerkin method with splines on a uniform mesh as test-trial functions. We also analyse a full discretization of the method based on the Galerkin collocation method using high order integration formulae to keep the optimal error estimates of the Galerkin method in weak norms. Asymptotic expansions of the error for this method are provided. Finally, we show how these expansions extend to the computation of the potential. The expansions of the error in powers of the discretization parameter are useful to obtain a posteriori estimates of the error and to apply Richardson extrapolation for acceleration of convergence. © 1999 Elsevier Science Ltd. All rights reserved.