MODIFIED PROJECTION AND THE ITERATED MODIFIED PROJECTION METHODS FOR NONLINEAR INTEGRAL EQUATIONS

摘要

Consider a nonlinear operator equation x ? K(x) = f, where K is a Urysohn integral operator with a smooth kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree ≤ r, previous authors have established an order r + 1 convergence for the Galerkin solution and 2r + 2 for the iterated Galerkin solution. Equivalent results have also been established for the interpolatory projection at Gauss points. In this paper, a modified projection method is shown to have convergence of order 3r + 3 and one step of iteration is shown to improve the order of convergence to 4r + 4. The size of the system of equations that must be solved, in implementing this method, remains the same as for the Galerkin method.