Application of the over-set grid technique to a model singular perturbation problem

The numerical solution of a singularly perturbed problem, in the form of a two-dimensional convection-diffusion equation, is studied by using the technique of over-set grids. For this purpose the Overture software library is used. The selection of component grids is made on basis of asymptotic analysis. The behavior of the solution is studied for a range of small diffusion parameters. Also the possibilities of rotating the grid with the convection direction is considered. In order to fit global properties of the solution, the composite grid used is made parameter dependent. In view of possible $eps$-uniform convergence, in the resulting composite grid the number of grid points is kept constant for the different values of the small parameter. Only the grid spacing is adapted, depending on the parameters. We see that, even with careful adaptation of the grid, no $eps$-uniform convergence is achieved.