An hp finite element method for 4th order singularly perturbed problems

Abstract

We present the analysis for the hp finite element approximation of the solution to singularly perturbed fourth order problems, using a balanced norm. In Panaseti et al. (2016) it was shown that the hp version of the Finite Element Method (FEM) on the so-called Spectral Boundary Layer Mesh yields robust exponential convergence when the error is measured in the natural energy norm associated with the problem. In the present article we sharpen the result by showing that the same hp-FEM on the Spectral Boundary Layer Mesh gives robust exponential convergence in a stronger, more balanced norm. As a corollary we also get robust exponential convergence in the maximum norm. The analysis is based on the ideas in Roos and Franz (Calcolo 51, 423---440, 2014) and Roos and Schopf (ZAMM 95, 551---565, 2015) and the recent results in Melenk and Xenophontos (2016). Numerical examples illustrating the theory are also presented.