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This paper presents an extended version of the article [Franz, S., Kopteva,N.: J. Differential Equations, 252 (2012)]. The main improvement compared tothe latter is in that here we additionally estimate the mixed second-orderderivative of the Green's function. The case of Neumann conditions along thecharacteristic boundaries is also addressed. A singularly perturbedconvection-diffusion problem is posed in the unit square with a horizontalconvective direction. Its solutions exhibit parabolic and exponential boundarylayers. Sharp estimates of the Green's function and its first- and second-orderderivatives are derived in the $L_1$ norm. The dependence of these estimates onthe small diffusion parameter is shown explicitly. The obtained estimates willbe used in a forthcoming numerical analysis of the considered problem.

Green's function estimates for a 2d singularly perturbed convection-diffusion problem: extended analysis