Generalised gradients for virtual elements and applications to a posteriori error analysis

TL;DR

This paper reformulates the virtual element method as a generalized gradient approach, enabling reliable a posteriori error estimation through local flux and potential reconstructions, with bounds independent of stabilization.

Contribution

It introduces a novel reformulation of virtual element methods as generalized gradients, facilitating error estimation with mesh-independent bounds.

Findings

Reliable error estimators with mesh-independent bounds

Effective local flux and potential reconstructions

Theoretical proof of upper and lower error bounds

Abstract

We rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials on elemental subtriangulations. We prove the usual upper and lower bounds with constants independent of the stabilisation of the method and, under technical assumptions on the mesh, the degree of accuracy.