Balanced norm estimates for $rp$-Finite Element Methods applied to singularly perturbed fourth order boundary value problems

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.