It has been reported that convergence stalls on Bakhvalov-Shishkin mesh in the case of $N^{-1}\le \epsilon$, where ε is the singular perturbation parameter and N is the number of mesh intervals. To analyze these phenomena, we present uniform convergence analysis of finite element methods on Bakhvalov-type meshes, one popular kind of graded meshes closely related to Bakhvalov-Shishkin mesh, when $N^{-1}\le \epsilon$. An optimal order of convergence is proved and this result is used for the improvement of Bakhvalov-Shishkin mesh. These theoretical results are verified by numerical experiments.

中文翻译：

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在的情况下，上Bakhvalov型网格的有限元方法一致收敛Ñ ^-1  ≤  ε

据报道，在 ${ñ}^{-\mathrm{1个}}\le \epsilon$，其中ε是奇异摄动参数，N是网格间隔数。为了分析这些现象，我们对Bakhvalov型网格进行了有限元方法的统一收敛分析，Bakhvalov型网格是一种与Bakhvalov-Shishkin网格密切相关的流行渐变网格。${ñ}^{-\mathrm{1个}}\le \epsilon$。证明了收敛的最佳阶数，并将该结果用于改进Bakhvalov-Shishkin网格。这些理论结果通过数值实验得到了验证。