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

TL;DR

This paper provides a rigorous error analysis for a finite element method on a layer-adapted mesh applied to a 2D turning point problem, ensuring uniform convergence and robustness in capturing sharp layers.

Contribution

It introduces a uniform error estimate in the maximum norm for a finite element approximation on a Shishkin mesh for a 2D turning point problem.

Findings

Proves uniform convergence in x-layer regions.

Establishes ε-independent bounds in coarse regions.

Demonstrates robustness in capturing sharp solution layers.

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 $\varepsilon$-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.