A fast numerical solution method for two dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation

In this paper we consider fast numerical solution methods for two dimensional Fredholm integral equation of the second kindf(x,y)-∫αβ∫αβa(x,y,u,v)f(u,v)dudv=g(x,y),(x,y)∈[α,β]×[α,β],where a(x, y, u, v) is smooth and g(x, y) is in L2[α, β]2. Discretizing the integral equation by certain quadrature rule, we get a linear system. To deduce fast approximate solution methods for the resulted linear system, we study the approximation of the four-variable kernel function a(x, y, u, v) by piecewise polynomial: partition the domain [α, β]4 into subdomains of the same size and interpolate the kernel function a(x, y, u, v) in each subdomain. Fast matrix-vector multiplication algorithms and efficient iterative methods are derived. Numerical results are given to illustrate the efficiency of our methods.