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

12/22/2022

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by   Sebastian Franz, et al.

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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 to the latter is in that here we additionally estimate the mixed second-order derivative of the Green's function. The case of Neumann conditions along the characteristic boundaries is also addressed. A singularly perturbed convection-diffusion problem is posed in the unit square with a horizontal convective direction. Its solutions exhibit parabolic and exponential boundary layers. Sharp estimates of the Green's function and its first- and second-order derivatives are derived in the L_1 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.