hp Finite Element Methods for Fourth Order Singularly Perturbed Boundary Value Problems

We consider fourth order singularly perturbed boundary value problems (BVPs) in one-dimension and the approximation of their solution by the hp version of the Finite Element Method (FEM). If the given problem’s boundary conditions are suitable for writing the BVP as a second order system, then we construct an hp FEM on the so-called Spectral Boundary Layer Mesh that gives a robust approximation that converges exponentially in the energy norm, provided the data of the problem is analytic. We also consider the case when the BVP is not written as a second order system and the approximation belongs to a finite dimensional subspace of the Sobolev space H. For this case we construct suitable C1−conforming hierarchical basis functions for the approximation and we again illustrate that the hp FEM on the Spectral Boundary Layer Mesh yields a robust approximation that converges exponentially. A numerical example that validates the theory is also presented.