Adaptive computation for convection dominated diffusion problems

We derive sharp L∞(L ¹ ) a posteriori error estimate for the convection dominated diffusion equations of the form

$$frac{{partial u}}{{partial t}} + div(vu) - varepsilon Delta u = g.$$

The derived estimate is insensitive to the diffusion parameter ε → 0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements. Numerical experiments are reported to show the competitive behavior of the proposed adaptive method.