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.

翻译：在Bakhvalov型网格上，针对具有指数层的二维奇异摄动对流扩散问题，有限元方法的一致收敛性分析仍是一个开放性问题。先前的尝试均未成功。主要挑战在于：层内靠近过渡点的网格子域宽度、狄利克雷边界条件的限制以及指数层的结构特性。为应对这些挑战，本文首次提出了一种新型分析技术，该技术充分利用了插值特性以及边界上光滑函数与层函数之间的关联。结合该技术与一种结构简单的新型插值方法，可证明任意阶k的有限元方法在能量范数下具有最优阶k+1的一致收敛性。数值实验验证了我们的理论结果。