The convergence of classical Schwarz methods applied to convection–diffusion problems with regular boundary layers Original Research Article

The problem of constructing a parameter-uniform iterative numerical method based on a classical Schwarz alternating procedure and the principles of the Shishkin fitted mesh is considered. It is first shown that an overlapping Schwarz method using uniform meshes and arbitrary fixed interface positions produces numerical approximations which are not ε-uniform convergent. A numerical method using Shishkin interface positions and uniform meshes on overlapping meshes also produces approximations which do not converge to the true solution. Finally, a non-overlapping method using Shishkin interface positions, uniform meshes and artificial Dirichlet interface conditions is examined for a two-dimensional elliptic problem with regular boundary layers and it is shown to be essentially first order convergent for ε⩽N−1.