Why Classical Finite difference Approximations fail for a singularly perturbed System of Convection-Diffusion Equations

Abstract

We consider classical Finite Difference Scheme for a system of singularly perturbed convection-diffusion equations coupled in their reactive terms, we prove that the classical SFD scheme is not a robust technique for solving such problem with singularities. First we prove that the discrete operator satisfies a stability property in the l 2 -norm which is not uniform with respect to the perturbation parameters, as the solution blows up when the perturbation parameters goes to zero. An error analysis also shows that the solution of the SFD is not uniformly convergent in the discrete l 2 -norm with respect to the perturbation parameters, i.e., the convergence is very poor for a sufficiently small choice of the perturbation parameters. Finally we present numerical results that confirm our theoretical findings.