Convergence of Higher Order Finite Element Method on eXp-Bakhvalov Mesh for a 2D Convection-Diffusion Equation with Exponential Boundary Layers

Abstract Roos and Stynes [Some open questions in the numerical analysis of singularly perturbed differential equations, Comput. Methods Appl. Math. 15 2015, 4, 531–550] posed several open problems, one of which is obtaining uniform convergence of higher-order finite element methods on Bakhvalov-type meshes. In this article, we propose a finite element analysis of any order on an eXp-Bakhvalov mesh to solve a two-dimensional singularly perturbed boundary value problem, whose solution exhibits exponential layers. A careful selection of the interpolation operator, considering the characteristics of the layers, allows the finite element method to achieve optimal-order convergence with respect to the singular perturbation parameter. Numerical results are presented to support the theoretical findings.