On the sharpness of Green's function estimates for a convection-diffusion problem
This work provides rigorous theoretical validation of existing estimates for a specific class of singularly perturbed PDEs, which is incremental for researchers in numerical analysis.
The authors prove the sharpness of their previously derived upper bounds for Green's function and its derivatives in the L1 norm for a 3D convection-diffusion problem with characteristic layers, by establishing matching lower bounds that explicitly depend on the singular perturbation parameter.
Linear singularly perturbed convection-diffusion problems with characteristic layers are considered in three dimensions. We demonstrate the sharpness of our recently obtained upper bounds for the associated Green's function and its derivatives in the $L_1$ norm. For this, in this paper we establish the corresponding lower bounds. Both upper and lower bounds explicitly show any dependence on the singular perturbation parameter.