MONOTONE RELAXATION ITERATES AND APPLICATIONS TO SEMILINEAR SINGULARLY PERTURBED PROBLEMS

Abstract. This paper deals with monotone relaxation iterates for solving nonlinear monotone dif-ference schemes of elliptic type. The monotone ω-Jacobi and SUR (Successive Under-Relaxation) methods are constructed. The monotone methods solve only linear discrete systems at each iter-ative step and converge monotonically to the exact solution of the nonlinear monotone difference schemes. Convergent rates of the monotone methods are estimated. The proposed methods are applied to solving semilinear singularly perturbed reaction-diffusion problems. Uniform conver-gence of the monotone methods is proved. Numerical experiments complement the theoretical results. Key words. semilinear elliptic problem, monotone difference schemes, monotone relaxation iterates, singularly perturbed problems, uniform convergence. 1