Optimal error analysis of an interior penalty virtual element method for fourth-order singular perturbation problems
Fang Feng, Yuanyi Sun, Yue Yu
TL;DR
This work addresses the challenge of obtaining optimal, uniform error estimates for the Interior Penalty Virtual Element Method (IPVEM) applied to the fourth-order singular perturbation problem $\varepsilon^2\Delta^2 u-\Delta u=f$ with $0<\varepsilon\le1$. By leveraging a Nitsche-type stabilization and projection-based enhancements through the elliptic projections $\Pi_K^{\nabla}$ and $\Pi_K^{\Delta}$, the IPVEM achieves an optimal, uniform error bound in the energy-like norm $\|\cdot\|_{\varepsilon,h}$, even in the presence of boundary layers. The analysis yields $\|u-u_h\|_{\varepsilon,h} \lesssim \varepsilon^{1/2}\|f\| + h^{m-1}|u^0|_m + h^k|f|_{k-1}$ for suitable regularity, and in the lowest-order case ($k=2$) provides a second-order uniform convergence when $f\in H^{1}(\Omega)$. Numerical experiments on unit square and L-shaped domains confirm the theoretical rates and demonstrate robustness to boundary layers, with the implementation available in the mVEM MATLAB package.
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.