AbstractWe consider fourth order singularly perturbed problems in one-dimension and
the approximation of their solution by the h version of the finite element method. In particular, we use piecewise Hermite polynomials of degree ${p\geq 3}$
defined on an exponentially graded mesh. We show that the
method converges uniformly, with respect to the singular perturbation
parameter, at the optimal rate when the error is measured in both the energy norm
and a stronger, ‘balanced’ norm. Finally, we illustrate our theoretical findings
through numerical computations, including a comparison with another scheme from the literature.