A superconvergence result for solutions of compact operator equations

Over the last 20 years, since the publication of Sloan's paper on the improvement by the iteration technique, vaxious approaches have been proposed for post-processing the Galerkin solution of multi-dimensional second kind Fredholm Integral equation. These methods include the iterated Galerkin method proposed by Sloan, the Kantorovich method and the iterated Kantorovich method. Recently, Lin, Zhang and Yan have proposed interpolation as an alternative to the iteration technique. For an integral operator, with a smooth kernel using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree less than or equal to r-1, previous authors have established an order r convergence for the Galerkin solution and 2r for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points and some interpolation post-processing technique. In this paper, a method is introduced and shown to have convergence of order 4r. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.