Optimal error analysis of an interior penalty virtual element method for fourth-order singular perturbation problems

TL;DR

This paper proves that an interior penalty virtual element method achieves optimal, uniform error estimates for fourth-order singular perturbation problems, improving upon previous suboptimal convergence rates.

Contribution

The paper establishes the optimal and uniform error estimates of IPVEM for singular perturbation problems, even with boundary layers, supported by numerical validation.

Findings

IPVEM achieves optimal, uniform error estimates.

Numerical experiments confirm theoretical error bounds.

Method effectively handles boundary layers in singular perturbation problems.

Abstract

In recent studies \cite{ZZ24, FY24}, the Interior Penalty Virtual Element Method (IPVEM) has been developed for solving a fourth-order singular perturbation problem, with uniform convergence established in the lowest-order case concerning the perturbation parameter. However, the resulting uniform convergence rate is only of half-order, which is suboptimal. In this work, we demonstrate that the proposed IPVEM in fact achieves optimal and uniform error estimates, even in the presence of boundary layers. The theoretical results are substantiated through extensive numerical experiments, which confirm the validity of the error estimates and highlight the method's effectiveness for singularly perturbed problems.