Accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems

An accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented. The mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite-difference or finite-element method. It is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions. This result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions. An application is given to a tabular reactor model from chemical engineering for numerical solutions, and the number of iterations are compared with that by the regular monotone iterative scheme.