Collocation methods for solving multivariable integral equations of the second kind

In a recent paper (Allouch, in press) [5] on one dimensional integral equations of the second kind, we have introduced new collocation methods. These methods are based on an interpolatory projection at Gauss points onto a space of discontinuous piecewise polynomials of degree rr which are inspired by Kulkarni’s methods (Kulkarni, 2003) [10], and have been shown to give a 4r+44r+4 convergence for suitable smooth kernels. In this paper, these methods are extended to multi-dimensional second kind equations and are shown to have a convergence of order 2r+42r+4. The size of the systems of equations that must be solved in implementing these methods remains the same as for Kulkarni’s methods. A two-grid iteration convergent method for solving the system of equations based on these new methods is also defined.