Greenʼs function estimates for a singularly perturbed convection–diffusion problem

We consider a singularly perturbed convection–diffusion problem posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Greenʼs function and its first- and second-order derivatives are derived in the L1 norm. The dependence of these estimates on the small diffusion parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem.