An Accurate Numerical Solution Of A Two Dimensional Heat Transfer Problem With A Parabolic Boundary Layer
C. Clavero 1, J. J. H. Miller 2, E. O'Riordan 3, G. I. Shishkin 41 Dpto. de Matematica Aplicada, Universidad de Zaragoza, Zaragoza,Spain
2 Mathematics Department, Trinity College, Dublin 2, Ireland
3 School of Mathematical Sciences, Dublin City University, Dublin 9,Ireland
4 Institute of Mathematics and Mechanics, Ekaterinburg, Russia
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.