Convergence of a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection--diffusion equation in 2D

TL;DR

This paper demonstrates that a finite element method applied on a Bakhvalov-type mesh achieves uniform, nearly optimal convergence for solving 2D singularly perturbed convection-diffusion equations with boundary layers.

Contribution

It introduces a finite element approach on a Bakhvalov-type mesh with a proof of uniform convergence for 2D problems exhibiting boundary layers.

Findings

Proves uniform convergence of the method.

Achieves almost optimal order of accuracy.

Handles exponential boundary layers effectively.

Abstract

A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection--diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.