Numerical simulations of boundary layer problems

At the interface between two media there often appear boundary layers. Singularly perturbed Helmholz equation is typical example. Up-to-date finite difference methods are shown to be capable of effective solving of such problems. Convergence verification procedure is proposed that does not require a priori estimations construction. A superfast algorithm that provides a posteriori asymptotically precise error estimation is described and semi-uniform rectangular grid that resolves all parts of solution is proposed. The algorithm proposed makes it possible to achieve good precisions on moderate grids with number of points $N\sim 200$ in each direction. This algorithm is realized as a program in Matlab environment.