Finite element method for singularly perturbed problems with two parameters on a Bakhvalov-type mesh in 2D

TL;DR

This paper develops and analyzes finite element methods of any order for a 2D singularly perturbed elliptic problem with two parameters, using a Bakhvalov-type mesh to achieve optimal convergence.

Contribution

It introduces a new interpolation technique and a refined mesh analysis to prove optimal convergence of finite element methods for problems with two small parameters.

Findings

Achieved optimal convergence order for the finite element methods.

Developed a new interpolation method based on layer characteristics.

Provided a detailed analysis of mesh scale near exponential layers.

Abstract

For a singularly perturbed elliptic model problem with two small parameters, we analyze finite element methods of any order on a Bakhvalov-type mesh. For convergence analysis, we construct a new interpolation by using the characteristics of layers. Besides, a more subtle analysis of the mesh scale near the exponential layer is carried out. Based on the interpolation and new analysis of the mesh scale, we prove the optimal convergence order.