Convergence of finite difference methods for convection–diffusion problems with singular solutions

We are concerned with initial-boundary value problems of convection–diffusion equations in a square, whose solutions have unbounded derivatives near the boundary. By using finite difference approximations with respect to spatial variables and an implicit method with respect to the time variable, it is shown that the numerical solution is convergent if the derivatives go to infinity under proper conditions. Furthermore, the convergence of numerical solution can be accelerated if the mesh points are some functions of equidistant mesh points.