For singularly perturbed convection-diffusion problems, supercloseness analysis of finite element method is still open on Bakhvalov-type meshes, especially in the case of 2D. The difficulties arise from the width of the mesh in the layer adjacent to the transition point, resulting in a suboptimal estimate for convergence. Existing analysis techniques cannot handle these difficulties well. To fill this gap, a novel interpolation is designed delicately for the first time for the smooth part of the solution, bringing about the optimal supercloseness result of almost order 2 under an energy norm for finite element method. Our theoretical result is uniformly in the singular perturbation parameter and is supported by the numerical experiments.

翻译：对于奇异摄动对流扩散问题，在Bakhvalov型网格上（尤其是二维情形）的有限元超逼近分析仍属未解决难题。困难源于层内过渡点附近网格宽度导致收敛估计次优，现有分析技术难以有效应对。为填补这一空白，本文首次针对解的光滑部分创新设计了一种新型插值格式，使得有限元方法在能量范数下获得了几乎二阶的最优超逼近结果。该理论结果关于奇异摄动参数一致成立，并通过数值实验得到验证。