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

On a Bakhvalov-type mesh widely used for boundary layers, we consider the finite element method for singularly perturbed elliptic problems with two parameters on the unit square. It is a very challenging task to analyze uniform convergence of finite element method on this mesh in 2D. The existing analysis tool, quasi-interpolation, is only applicable to one-dimensional case because of the complexity of Bakhvalov-type mesh in 2D. In this paper, a powerful tool, Lagrange-type interpolation, is proposed, which is simple and effective and can be used in both 1D and 2D. The application of this interpolation in 2D must be handled carefully. Some boundary correction terms must be introduced to maintain the homogeneous Dirichlet boundary condition. These correction terms are difficult to be handled because the traditional analysis do not work for them. To overcome this difficulty, we derive a delicate estimation of the width of some mesh. Moreover, we adopt different analysis strategies for different layers. Finally, we prove uniform convergence of optimal order. Numerical results verify the theoretical analysis.