A Naive Finite difference Approximations for Singularly Perturbed Parabolic Reaction-Diffusion problems

Abstract

In this thesis, we treated a Standard Finite Difference Scheme for a singularly perturbed parabolic reaction-diffusion equation. We proved that the Standard Finite Difference Scheme is not a robust technique for solving such problems with singularities. First we discretized the continuous problem in time using the forward Euler method. We proved that the discrete problem satisfied a stability property in the l ∞ − norm and l 2 − norm which is not uniform with respect to the perturbation parameter, as the solution is unbounded when the perturbation parameter goes to zero. Error analysis also showed that the solution of the SFDS is not uniformly convergent in the discrete l ∞ − norm with respect to the perturbation parameter, (i.e., the convergence is very poor as the parameter becomes very small). Finally we presented numerical results that confirmed our theoretical findings.