A posteriori error estimates for a $C^1$ virtual element method applied to the thin plate vibration problem

TL;DR

This paper develops residual-based a posteriori error estimates for a conforming $C^1$ virtual element method applied to thin plate vibration problems in 2D and 3D, with proven reliability and efficiency.

Contribution

It introduces a novel a posteriori error estimator for $C^1$ virtual elements on polygonal and polyhedral meshes, with dimension-independent reliability and efficiency proofs.

Findings

Error estimator is reliable and efficient in 2D and 3D.

Numerical experiments confirm optimal performance.

Method is applicable to complex polygonal and polyhedral meshes.

Abstract

We propose and analyse residual-based a posteriori error estimates for the virtual element discretisation applied to the thin plate vibration problem in both two and three dimensions. Our approach involves a conforming $C^1$ discrete formulation suitable for meshes composed of polygons and polyhedra. The reliability and efficiency of the error estimator are established through a dimension-independent proof. Finally, several numerical experiments are reported to demonstrate the optimal performance of the method in 2D and 3D.