Supercloseness of finite element method on a Bakhvalov-type mesh for a singularly perturbed problem with two parameters

TL;DR

This paper investigates the supercloseness properties of the finite element method on a Bakhvalov-type mesh for a two-parameter singularly perturbed problem with exponential boundary layers, introducing a new interpolation technique.

Contribution

It introduces a simple new interpolation method for convergence analysis and reveals a subtle relationship between the mesh and the exponential layers, leading to supercloseness results.

Findings

Supercloseness between Lagrange interpolation and numerical solution proven.

Numerical tests confirm theoretical supercloseness results.

New interpolation simplifies convergence analysis for singularly perturbed problems.

Abstract

In this paper, the linear finite element method on a Bakhvalov-type mesh is applied to a singularly perturbed problem with two parameters. The solution of the problem exists two exponential boundary layers. A new interpolation, which is simple in construction and analysis, is introduced for convergence analysis. Furthermore, we find a subtle relationship between the Bakhvalov-type mesh itself and the weaker exponential layer and obtain an interesting result. Finally, we prove a supercloseness result between the Lagrange interpolation and the numerical solution. Numerical tests confirm our theoretical results.