ON THE SINGULARLY PERTURBED SEMILINEAR REACTION-DIFFUSION PROBLEM AND ITS NUMERICAL SOLUTION

Abstract. We obtain improved derivative estimates for the solution of the semilinear singularly perturbed reaction-diffusion problem in one dimension. This enables us to modify the transition points between the fine and coarse parts of the Shishkin discretization mesh. We prove that the numerical solution, obtained by using the central finite-difference scheme on the modified mesh, retains the same order of convergence uniform in the perturbation parameter as on the standard Shishkin mesh. However, the modified mesh may be denser in the layers than the standard one, and, when this is the case, numerical results show an improvement in the accuracy of the computed solution. Key words. singularly perturbed boundary-value problem, reaction-diffusion, Shishkin mesh, finite differences, and uniform convergence