Optimal error analysis of an interior penalty virtual element method for fourth-order singular perturbation problems
This work provides improved error analysis for a numerical method addressing singular perturbation problems, which is incremental but important for computational mathematics.
The authors tackled the suboptimal convergence rate of the Interior Penalty Virtual Element Method for fourth-order singular perturbation problems, demonstrating that it achieves optimal and uniform error estimates even with boundary layers, as confirmed by numerical experiments.
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.