A C1-conforming hp finite element method for fourth order singularly perturbed boundary value problems

We consider a fourth order singularly perturbed boundary value problem (BVP) in one-dimension and the approximation of its solution by the hp   version of the Finite Element Method (FEM). The given problem's boundary conditions are not suitable for writing the BVP as a second order system, hence the approximation will be sought from a finite dimensional subspace of the Sobolev space H2H2. We construct suitable C1C1 hierarchical basis functions for the approximation and we show that the hp FEM on the Spectral Boundary Layer Mesh yields a robust approximation that converges exponentially in the energy norm, as the number of degrees of freedom is increased. Numerical examples that validate (and extend) the theory are also presented.