Accelerating fixed point algorithms for nonlinear Fredholm integral equations

This paper proposes a simple idea to speed-up the convergence of a fixed-point iteration for Fredholm equation defined on a mesh. First, the analytical problem is discretized by using quadrature rule and collocating at mesh points. Banach Fixed-Point Theorem is used to construct a discrete Picard scheme (PS). To accelerate the computations, the Picard scheme is modified to form a new faster algorithm christened “Gauss-Seidel-Picard” (GSP) scheme. The GSP scheme modifies the classical Picard iteration in the same way as the Gauss-Seidel iteration modifies the Jacobi iteration. Numerous questions arise, namely: does the GSP scheme converge? If it does converge, is it truly better (faster) than the classical Picard scheme? To address these questions, the convergence errors of the two schemes are derived and it is shown that the GSP scheme converges faster than the Picard scheme. Several numerical experiments are provided to verify this claim in terms of both number of iterations and CPU time. It is found that the GSP solver is much more computationally efficient than the classical scheme. In particular, for the examples considered in the work, the GSP scheme is between 30% to 48% more efficient than the standard Picard scheme. It is suggested that this idea be extended to other fixed point methods and to other problems, even in higher dimensions.