A posteriori error estimation and adaptivity in $hp$ virtual elements

TL;DR

This paper introduces a new a posteriori error estimator for the $hp$ virtual element method, enabling adaptive mesh refinement on complex polygonal meshes, validated through numerical experiments.

Contribution

It presents a novel explicit error estimator with bounds for VEM, along with an $hp$ Clément quasi-interpolant, advancing adaptive methods for polygonal meshes.

Findings

The error estimator provides reliable bounds for the energy error.

Adaptive $hp$ refinement improves solution accuracy on complex meshes.

Numerical experiments confirm the effectiveness of the proposed methods.

Abstract

An explicit and computable error estimator for the $hp$ version of the virtual element method (VEM), together with lower and upper bounds with respect to the exact energy error, is presented. Such error estimator is employed to provide $hp$ adaptive mesh refinements for very general polygonal meshes. In addition, a novel VEM $hp$ Cl\'ement quasi-interpolant, instrumental for the a posteriori error analysis, is introduced. The performances of the adaptive method are validated by a number of numerical experiments.