Full analysis of the Green's function for a singularly perturbed convection-diffusion problem in three dimensions

TL;DR

This paper derives sharp bounds for the Green's function of a 3D singularly perturbed convection-diffusion problem, explicitly showing how these bounds depend on the small perturbation parameter, aiding future numerical analysis.

Contribution

It provides detailed bounds for the Green's function and its derivatives in three dimensions, explicitly relating them to the perturbation parameter, which was not previously established.

Findings

Established sharp $L_1$ bounds for Green's function and derivatives.

Explicitly demonstrated dependence on the perturbation parameter.

Facilitates future numerical analysis of the problem.

Abstract

A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its derivatives are established in the $L_1$ norm. The dependence of these bounds on the small perturbation parameter is shown explicitly. The obtained estimates will be used in a forthcoming numerical analysis of the considered problem. The present article is a more detailed version of our recent paper [7].