A posteriori analysis of a virtual element approach on polytopal meshes for the buckling eigenvalue problem

TL;DR

This paper develops and validates a residual-based a posteriori error estimator for the conforming C^1 Virtual Element Method applied to buckling eigenvalue problems, including nonlinear effects, on general polyhedral meshes.

Contribution

It introduces a fully computable, reliable, and efficient error estimator for VEM in buckling problems with nonlinear effects on complex meshes.

Findings

Estimator achieves optimal accuracy in 2D and 3D

Robustness demonstrated across various mesh types

Implementation available in open-source vem++ library

Abstract

We introduce a novel residual-based a posteriori error estimator for the conforming $C^1$ Virtual Element Method (VEM) applied to the buckling eigenvalue problem, incorporating nonlinear plane stress effects in both two and three dimensions. The estimator is fully computable on general polyhedral meshes and implemented within the open-source \texttt{vem++} library. Its reliability is rigorously justified via bounds on the residual equation using polynomial projections, stabilisation contributions, and interpolation estimates, while efficiency is ensured through the use of bubble function arguments. Comprehensive numerical experiments in 2D and 3D illustrate the estimator's optimal accuracy and robustness, highlighting its potential for predictive analysis of complex plate structures.