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We consider fourth order singularly perturbed eigenvalue problems inone-dimension and the approximation of their solution by the $h$ version of theFinite Element Method (FEM). In particular, we use piecewise Hermitepolynomials of degree $p\geq 3$ defined on an {\emph{exponentially graded}}mesh. We show that the method converges uniformly, with respect to the singularperturbation parameter, at the optimal rate when the error in the eigenvaluesis measured in absolute value and the error in the eigenvectors is measured inthe energy norm. We also illustrate our theoretical findings through numericalcomputations for the case $p=3$.

On the finite element approximation of fourth order singularly perturbed eigenvalue problems