In convergence analysis of finite element methods for singularly perturbed reaction–diffusion problems, balanced norms have been successfully introduced to replace standard energy norms so that layers can be captured. In this article, we focus on the convergence analysis in a balanced norm on Bakhvalov-type rectangular meshes. In order to achieve our goal, a novel interpolation operator, which consists of a local \(L^2\) projection operator and the Lagrange interpolation operator, is introduced for a convergence analysis of optimal order in the balanced norm. The analysis also depends on the stabilities of the \(L^2\) projection and the characteristics of Bakhvalov-type meshes. Furthermore, we obtain a supercloseness result in the balanced norm, which appears in the literature for the first time. This result depends on another novel interpolant, which consists of the local \(L^2\) projection operator, a vertices-edges-element operator and some corrections on the boundary.

在奇异扰动反应扩散问题的有限元方法的收敛分析中，已成功引入平衡范数来代替标准能量范数，以便可以捕获层。在本文中，我们重点讨论 Bakhvalov 型矩形网格的平衡范数收敛分析。为了实现我们的目标，引入了一种由局部\(L^2\)投影算子和拉格朗日插值算子组成的新型插值算子，用于平衡范数中最优阶次的收敛分析。分析还取决于\(L^2\)Bakhvalov 型网格的投影和特征。此外，我们在平衡范数中获得了超接近性结果，这是首次出现在文献中。这个结果取决于另一个新的插值，它由局部\(L^2\)投影算子、一个顶点-边-元素算子和一些边界修正组成。