Numerical solution of nonlinear elliptic systems by block monotone iterations

Abstract

This paper deals with investigating numerical methods for solving coupled system of nonlinear parabolic problems. We utilize block monotone iterative methods based on Jacobi and Gauss-Seidel methods to solve difference schemes which approximate the coupled system of nonlinear parabolic problems, where reaction functions are quasimonotone nondecreasing or nonincreasing. In the view of upper and lower solutions method, two monotone upper and lower sequences of solutions are constructed, where the monotone property ensures the theorem on existence of solutions to problems with quasi-monotone nondecreasing and nonincreasing reaction functions. Construction of initial upper and lower solutions is presented. The sequences of solutions generated by the block Gauss-Seidel method converge not slower than by the block Jacobi method.