A C~1-conforming hp finite element method for fourth order singularly perturbed boundary value problems

摘要

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 H~2. We construct suitable C~1 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.