Survey

Here we sketch the state in the development of special difference schemes for, in my thinking, “advanced” problems with boundary layers. First, we discuss some applications of special finite difference schemes to solve problems with boundary layers. Next, we outline the use of special difference schemes to approximate boundary value problems for parabolic equations with piecewise-smooth and discontinuous initial-boundary conditions. Approximations of derivatives are also discussed. We touch on an approach in the construction of difference schemes based on a posteriori adaptive meshes. For an elliptic problem in an unbounded domain, we consider an approach to construct difference schemes on meshes with a finite number of mesh nodes whose solutions converge ε-uniformly on prescribed bounded subdomains. For an elliptic problem in a rectangle for a convection-diffusion equation with the perturbation vector-parameter ε, we give compatibility conditions that guarantee smoothness of the solution and its components which are required for the construction of ε-uniformly convergent schemes.