Layer-adapted Numerical Solution For Singularly Perturbed Convection-diffusion Problems

Singularly perturbed convection-diffusion problems involved in many branches of science and engineer. The classical numerical method does not give satisfactory results because the presence of perturbation parameters, especially when it is very small, how to solve the singular perturbation problems with numerical solution becomes a hot research topic, and attracts many domestic and foreign scholars to study it. From a different point of view, there are two methods mainly for dealing with singular perturbation problems with boundary layer now. One method is operator-adapted method which reflects the nature of the solution at the boundary layer. Another is the layer-adapted method. Combined with the current research status, we discuss the layer-adapted method in this paper, and we mainly do something as follows:(1)The first part is about singularly perturbed convection-diffusion Dirichlet boundary problems in one-dimension. First, we give the solution nature and Bakhvalov-Shishkin mesh, and we use the midpoint upwind difference scheme on Bakhvalov-Shishkin mesh to study the general situation. Second, its s-uniform convergence is proved by barrier functions and the comparison principle lemma. We get O(N-2) convergence rate in the coarse part, and O(N-1) in the fine part. Last, numerical example support our crror estimates.(2) In the second part, we study the Robin problem in one dimension. We discuss the midpoint upwind difference scheme on the Shishkin mesh to solve the Robin problems. We deal with the boundary condition in the right boundary by adding a virtual point. We get two order convergence rate in the coarse part, and almost one order convergence rate in the fine part. Several numerical examples support the elaborate error estimates..(3) The third part is about the two-dimensional problem. In this part, we put the problem to the general situation. We use the simple upwind scheme on the coarse part and central scheme on the fine part, and we get the one convergence rate on the whole mesh. We prove its ε-uniform convergence order by constructing barrier functions and using the comparison principle. The numerical experiments also support our theoretical result.