Fast and accurate solvers for weakly singular integral equations

Consider an integral equation lambda u - Tu = f, where T is an integral operator, defined on C[0, 1], with a kernel having an algebraic or a logarithmic singularity. Let pi(m) denote an interpolatory projection onto a space of piecewise polynomials of degree <= r - 1 with respect to a graded partition of [0, 1] consisting of m subintervals. In the product integration method, an approximate solution is obtained by solving lambda u(m) - T pi(m)u(m) = f. As in order to achieve a desired accuracy, one may have to choose m large, we find approximations of u(m) using a discrete modified projection method and its iterative version. We define a two-grid iteration scheme based on this method and show that it needs less number of iterates than the two-grid iteration scheme associated with the discrete collocation method. Numerical results are given which validate the theoretical results.