Morley-Wang-Xu element methods with penalty for a fourth order elliptic singular perturbation problem

TL;DR

This paper introduces two novel Morley-Wang-Xu element methods with penalty techniques for solving fourth order elliptic singular perturbation problems, providing robust error estimates and numerical validation.

Contribution

The paper develops two new penalty-based Morley-Wang-Xu element methods specifically designed for fourth order elliptic singular perturbation problems, with proven error estimates.

Findings

Robust a priori error estimates under minimal regularity.

Numerical results confirming theoretical error bounds.

Effective penalty formulations for improved stability.

Abstract

Two Morley-Wang-Xu element methods with penalty for the fourth order elliptic singular perturbation problem are proposed in this paper, including the interior penalty Morley-Wang-Xu element method and the super penalty Morley-Wang-Xu element method. The key idea in designing these two methods is combining the Morley-Wang-Xu element and penalty formulation for the Laplace operator. Robust a priori error estimates are derived under minimal regularity assumptions on the exact solution by means of some established a posteriori error estimates. Finally, we present some numerical results to demonstrate the theoretical estimates.