Abstract: In this paper, we discuss the singular perturbation problem of the parabolic partial differential equation. As usual, we must reduce the mesh size in the neighbourhood of the boundary layer so that typical feature of the boundary layer will not be lost. Then we need very large operational quantity when mesh sizes are getting too small.Now we propose the boundary layer scheme, which need not take very fine mesh size in the neighbourhood of the boundary layer. Numerical examples show that the accuracy can be satisfied with moderate step size.

Key words: mixed finite element method, shallow water equation, error estimate, current and silt sedimentation, characteristics method