A fast numerical solution method for two dimensional Fredholm integral equations of the second kind

In this paper we consider numerical solution methods for two dimensional Fredholm integral equation of the second kindf(x,y)−∫−11∫−11a(x,y,u,v)f(u,v)dudv=g(x,y),(x,y)∈[−1,1]×[−1,1], where a(x,y,u,v)a(x,y,u,v) is smooth and g(x,y)g(x,y) is in L2[−1,1]2L2[−1,1]2. We discuss polynomial interpolation methods for four-variable functions and then use the interpolating polynomial to approximate the kernel function a(x,y,u,v)a(x,y,u,v). Based on the approximation we deduce fast matrix-vector multiplication algorithms and efficient preconditioners for the above two dimensional integral equations. The residual correction scheme is used to solve the discretization linear system.