Convergence and supercloseness in a balanced norm of finite element methods on Bakhvalov-type meshes for reaction-diffusion problems

TL;DR

This paper establishes convergence and supercloseness results in a balanced norm for finite element methods on Bakhvalov-type meshes applied to reaction-diffusion problems, introducing novel interpolation operators for optimal analysis.

Contribution

It introduces new interpolation operators and provides the first supercloseness result in a balanced norm for these methods on Bakhvalov-type meshes.

Findings

Achieved optimal order convergence in the balanced norm.

Established the first supercloseness result in the literature.

Demonstrated stability of the proposed interpolation operators.

Abstract

In convergence analysis of finite element methods for singularly perturbed reaction--diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local weighted $L^2$ projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the $L^2$ projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local weighted $L^2$ projection operator, a vertices-edges-element operator and some corrections on the boundary.