Collocation methods using piecewise polynomials for second kind integral equations

Abstract

We consider the numerical solution of second kind Fredholm integral equations in one dimension by using the collocation method and its iterated variant. The collocation solution will be sought in a space of piecewise polynomials of order r. Superconvergence results for the iterated collocation solution are known when discontinuous piecewise polynomials are used and the collocation points are taken to be the r Gaussian points shifted to each subinterval. Here we give a corresponding superconvergence result for the iterated collocation solution when continuous piecewise polynomials with no continuity requirements on the derivatives are used. The collocation points are taken to be the knots plus the r−2 Lobatto points shifted to each subinterval. Some numerical results are also given.