The interior penalty virtual element method for fourth-order singular perturbation problems

TL;DR

This paper develops an interior penalty virtual element method for fourth-order singular perturbation problems, introducing modifications and an automated penalty parameter selection to achieve optimal and uniform convergence.

Contribution

It proposes a novel IPVEM with modifications and automated penalty selection, ensuring optimal and uniform convergence for singular perturbation problems.

Findings

Achieves optimal convergence in energy norm.

Proves uniform convergence with respect to the perturbation parameter.

Introduces modifications to penalty terms and automated parameter selection.

Abstract

This paper is dedicated to the numerical solution of a fourth-order singular perturbation problem using the interior penalty virtual element method (IPVEM) proposed in [42]. The study introduces modifications to the jumps and averages in the penalty term, as well as presents an automated mesh-dependent selection of the penalty parameter. Drawing inspiration from the modified Morley finite element methods, we leverage the conforming interpolation technique to handle the lower part of the bilinear form. Through our analysis, we establish optimal convergence in the energy norm and provide a rigorous proof of uniform convergence concerning the perturbation parameter in the lowest-order case.