We establish robust exponential convergence for $rp$-Finite Element Methods (FEMs) applied to fourth order singularly perturbed boundary value problems, in a \emph{balanced norm} which is stronger than the usual energy norm associated with the problem. As a corollary, we get robust exponential convergence in the maximum norm. $r p$ FEMs are simply $p$ FEMs with possible repositioning of the (fixed number of) nodes. This is done for a $C^1$ Galerkin FEM in 1-D, and a $C^0$ mixed FEM in 2-D over domains with smooth boundary. In both cases we utilize the \emph{Spectral Boundary Layer} mesh.

翻译：我们建立了四阶奇异摄动边值问题的$rp$-有限元方法（FEMs）在一种\emph{平衡范数}下的鲁棒指数收敛性，该范数比问题通常相关的能量范数更强。作为推论，我们在最大范数下获得了鲁棒指数收敛性。$rp$有限元方法本质上是允许（固定数量的）节点重新定位的$p$有限元方法。这针对一维中的$C^1$ Galerkin有限元法和二维中具有光滑边界区域上的$C^0$混合有限元法实现。在两种情况下，我们都利用了\emph{谱边界层}网格。