Solution of singularly perturbed convection–diffusion problems by the local Green's function method

As applied to one-dimensional singularly perturbed problems, methods based on local Green's functions exhibit fast convergence and numerical stability even if the problems involve sharp boundary layers. However, such methods virtually were not applied to problems in two or more space variables, since local Green's functions cannot be derived in a closed analytical form in these cases. In the present paper two-dimensional convection–diffusion problems are used as an example to show high efficiency of the Petrov–Galerkin discretization scheme in which the load Green's functions are used as projectors. The Green's functions are constructed on the basis of semianalytical integral representations proposed earlier. Asymptotic expansions of the Green's functions are also derived. They remove the existing limits of the practical applicability of the method with respect to the singularity parameter tending to zero. Test comparisons and numerical examples for an inhomogeneous convection field demonstrate stability of the solutions with minimal costs, which stabilize as $\varepsilon\to 0$.