A Superconvergence Result for Approximate Solution of Integral Equations of the Second Kind

Over the last 20 years, various approaches have been proposed for post-processing the Galerkin solution of second kind Fredholm Integral equation. These methods include the iterated Galerkin method proposed by Sloan, the Kantorovich method and the iterated Kantorovich method. For an integral operator with a smooth kernel, using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree $\leq r-1$, 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. In this talk, a recently proposed method by the author will be shown to have the 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. Similar results hold for approximate solution of eigenvalue problem.