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

TL;DR

This paper analyzes the finite element method's effectiveness in approximating eigenvalues and eigenvectors of fourth order singularly perturbed problems in one dimension, demonstrating uniform convergence using exponentially graded meshes.

Contribution

It introduces a finite element approach with Hermite polynomials on exponentially graded meshes that achieves uniform convergence for singularly perturbed eigenvalue problems.

Findings

Method converges uniformly at optimal rate

Eigenvalue errors measured in absolute value

Eigenvector errors measured in energy norm

Abstract

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