Optimal order of uniform convergence for finite element method on Bakhvalov-type meshes

TL;DR

This paper introduces a new analysis for finite element methods on Bakhvalov-type meshes, achieving optimal uniform convergence order for singularly perturbed boundary value problems, supported by numerical validation.

Contribution

A novel interpolant and analysis method that establish optimal uniform convergence order for finite element solutions on Bakhvalov-type meshes.

Findings

Proved optimal order of uniform convergence.

Validated theoretical results with numerical experiments.

Abstract

We propose a new analysis of convergence for a $k$th order ($k\ge 1$) finite element method, which is applied on Bakhvalov-type meshes to a singularly perturbed two-point boundary value problem. A novel interpolant is introduced, which has a simple structure and is easy to generalize. By means of this interpolant, we prove an optimal order of uniform convergence with respect to the perturbation parameter. Numerical experiments illustrate these theoretical results.