  On Bakhvalov-type mesh, uniform convergence analysis of finite element method
for a 2-D singularly perturbed convection-diffusion problem with exponential
layers is still an open problem. Previous attempts have been unsuccessful. The
primary challenges are the width of the mesh subdomain in the layer adjacent to
the transition point, the restriction of the Dirichlet boundary condition, and
the structure of exponential layers. To address these challenges, a novel
analysis technique is introduced for the first time, which takes full advantage
of the characteristics of interpolation and the connection between the smooth
function and the layer function on the boundary. Utilizing this technique in
conjunction with a new interpolation featuring a simple structure, uniform
convergence of optimal order k+1 under an energy norm can be proven for finite
element method of any order k. Numerical experiments confirm our theoretical
results.
