An hp Finite Element Method for Fourth Order Singularly Perturbed Problems

Abstract

We present an hp Finite Element Method (FEM) for the approximation to the solution of singularly perturbed fourth order problems in one-dimension. In (Panaseti et al, Appl Numer Math 104:81–97, 2016) it was shown that the hp version of the FEM, on the so-called Spectral Boundary Layer Mesh (Melenk et al, IMA J Numer Anal 33(2):609–628, 2013) yields robust exponential convergence when the error is measured in the energy norm. This result is sharpened by showing that the same method gives robust exponential convergence in a stronger, more balanced norm. As a corollary, we also get exponential convergence in the maximum norm. A numerical example illustrating the theory is also presented.