C. Clavero ¹, J. J. H. Miller ², E. O'Riordan ³, G. I. Shishkin ⁴

A singularly perturbed linear convection-diffusion problem for heat transfer in two dimensions with a parabolic boundary layer is solved numerically. The numerical method consists of a special piecewise uniform mesh condensing in a neighbourhood of the parabolic layer and a standard finite difference operator satisfying a discrete maximum principle. The numerical computations demonstrate numerically that the method is $\va$-uniform in the sense that the rate of convergence and error constant of the method are independent of the singular perturbation parameter $\va$. This means that no matter how small the singular perturbation parameter $\va$ is, the numerical method produces solutions with guaranteed accuracy depending solely on the number of mesh points used.

Key words: Linear convection-diffusion; parabolic layer; piecewise uniform mesh; finite difference.