Uniformly stable rectangular elements for fourth order elliptic singular perturbation problems

TL;DR

This paper investigates rectangular finite element methods for fourth order elliptic singular perturbation problems, demonstrating uniform convergence of certain Morley elements and proposing an extended high order variant with confirmed theoretical and numerical results.

Contribution

It introduces a $C^0$ extended high order rectangular Morley element and proves its uniform convergence for singular perturbation problems.

Findings

Non-$C^0$ Morley element is uniformly convergent in energy norm.

Proposed $C^0$ extended high order Morley element also achieves uniform convergence.

Numerical experiments confirm theoretical convergence results.

Abstract

This paper analyzes rectangular finite element methods for fourth order elliptic singular perturbation problems. We show that the non-$C^0$ rectangular Morley element is uniformly convergent in the energy norm with respect to the perturbation parameter. We also propose a $C^0$ extended high order rectangular Morley element and prove the uniform convergence. Finally, we do some numerical experiments to confirm the theoretical results.