A balanced norm, rather than the common energy norm, is introduced to reflect the behavior of layers more accurately in the finite element method for singularly perturbed reaction–diffusion problems. Convergence of optimal order in the balanced norm has been proved in the case of rectangular finite elements. However, for triangular finite elements $P_k$ ($k\ge 2$), it is still open to prove this convergence result. With the help of the $L^2$-stability of a weighted $L^2$ projection, instead of the $L^{\infty }$-stability widely used in existing references, the geometric constraints on meshes are relaxed. As a result, the optimal order convergence in the balanced norm is proved in the case of Bakhvalov–Shishkin triangular meshes. Numerical experiments support theoretical results.

在奇异扰动反应扩散问题的有限元方法中，引入了平衡范数，而不是公共能量范数，以更准确地反映层的行为。在矩形有限元的情况下，证明了平衡范数中最优阶的收敛性。然而，对于三角形有限元${磷}_{克}$ ($克\ge 2$)，证明这个收敛结果仍然是开放的。在该组织的帮助下${升}^2$- 加权的稳定性 ${升}^2$ 投影，而不是 ${升}^{\infty }$-stability 在现有参考文献中广泛使用，网格上的几何约束被放宽。结果，在 Bakhvalov-Shishkin 三角网格的情况下证明了平衡范数中的最优阶收敛。数值实验支持理论结果。