Numerical simulations of boundary layer problems

Abstract

Boundary layers often appear at the interface of two media. A typical example is the singularly perturbed Helmholz equation. It is shown that the modern finite difference methods solve such problems with a high efficiency under an appropriate choice of the grid. A convergence verification procedure is proposed that does not require majorant estimation. A superfast algorithm yielding an a posteriori asymptotically precise error estimate is described and a quasi-uniform rectangular grid resolving in detail all the segments of the solution is proposed. The algorithm guarantees good precision even on the moderate grids having N ~ 200 points in each direction. This algorithm is implemented in MATLAB.