A posteriori error estimation for an interior penalty virtual element method of Kirchhoff plates

TL;DR

This paper develops a residual-type a posteriori error estimator for an interior penalty virtual element method applied to Kirchhoff plate bending problems, simplifying implementation and enabling effective adaptive mesh refinement.

Contribution

It introduces a novel a posteriori error estimator for IPVEM that incorporates the $H^1$-elliptic projector, improving robustness and efficiency in adaptive algorithms.

Findings

The error estimator is reliable and efficient.

Numerical tests confirm the robustness of the estimator.

Adaptive VEM with mesh refinement performs well.

Abstract

In this paper, we develop a residual-type a posteriori error estimation for an interior penalty virtual element method (IPVEM) for the Kirchhoff plate bending problem. Building on the work in \cite{FY2023IPVEM}, we adopt a modified discrete variational formulation that incorporates the $ H^1 $-elliptic projector in the jump and average terms. This allows us to simplify the numerical implementation by including the $ H^1 $-elliptic projector in the computable error estimators. We derive the reliability and efficiency of the a posteriori error bound by constructing an enriching operator and establishing some related error estimates that align with $C^0$-continuous interior penalty finite element methods. As observed in the a priori analysis, the interior penalty virtual elements exhibit similar behaviors to $C^0$-continuous elements despite its discontinuity. This observation extends to the a posteriori estimate since we do not need to account for the jumps of the function itself in the discrete scheme and the error estimators. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results from several benchmark tests confirm the robustness of the proposed error estimators and show the efficiency of the resulting adaptive VEM.