Accelerated monotone iterative methods for a boundary value problem of second-order discrete equations

An accelerated monotone iterative method for a boundary value problem of second-order discrete equations is presented. This method leads to an existence-comparison theorem as well as a computational algorithm for the solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Some numerical results are presented to illustrate the monotone convergence of the iterative sequences and the rate of convergence of the iterations.