Monotone relaxation iterates and applications to semilinear singularly perturbed problems (Q2894051)

Numerical methods for solving differential equations lead to solving system of difference equations, and most of these equations are nonlinear. Obtaining reliable and efficient methods for such equations is the theme in the course of solution. This paper extended the existing monotone iterative methods proposed in the paper [the author, Computing 78, No. 1, 17--30 (2006; Zbl 1104.65101)], to monotone relaxation methods of Jacobi- and Gauss-Seidel type iteration for solving nonlinear monotone difference schemes in the canonical form. The monotone \(\omega\)-Jacobi and Successive Under-Relaxation methods are constructed and then applied to solve singularly perturbed reaction-diffusion problem with Dirichlet boundary conditions. The proposed monotone relaxation methods is a valuable extension of existing monotone iterative method and also provide valuable source of idea for extending other existing iterative methods.