Approximate Solution of Multivariable Integral Equations of the Second Kind

Abstract

For a multidimensional integral equation of the second kind with a smooth kernel, using the orthogonal projection onto a space of discontinuous piecewise polynomials of degree r, Atkinson has established an order r + 1 convergence for the Galerkin solution and an order 2r+2 convergence for the iterated Galerkin solution. In a recent paper [15], a new method based on projections has been shown to give a 4r + 4 convergence for one-dimensional second kind integral equations. The size of the system of equations that must be solved in implementing this method remains the same as for the Galerkin method. In this paper, this method is extended to multi-dimensional second kind equations and is shown to have convergence of order 4r + 4. For interpolatory projections onto a space of piecewise polynomials, it is shown that the order of convergence of the new method improves on the previously established orders of convergence for the collocation and the iterated collocation methods. A two-grid norm convergent method based on the new method is also defined.