Uniform norm error estimate for rectangular finite element approximation of a 2D turning point problem

Abstract

This work presents error analysis for a finite element method applied to a two-dimensional singularly perturbed convection-diffusion turning point problem. Utilizing a layer-adapted Shishkin mesh, we prove uniform convergence in the maximum norm in the x-layer regions and $\epsilon$-independent bounds for the coarse region. The analysis, critically based on the properties of a discrete Green's function, guarantees the method's robustness and accuracy in capturing sharp solution layers.