Anisotropic a posteriori error estimate for the Virtual Element Method

TL;DR

This paper develops an anisotropic a posteriori error estimate for the Virtual Element Method, enabling adaptive mesh refinement that improves efficiency and accuracy in solving 2D elliptic problems.

Contribution

It introduces a novel anisotropic error estimator and an adaptive polygonal algorithm, demonstrating enhanced performance over isotropic refinement methods.

Findings

The anisotropic estimator is reliable for VEM approximations.

Adaptive anisotropic refinement outperforms isotropic schemes.

Numerical tests confirm improved accuracy and efficiency.

Abstract

We derive an anisotropic a posteriori error estimate for the adaptive conforming Virtual Element approximation of a paradigmatic two-dimensional elliptic problem. In particular, we introduce a quasi-interpolant operator and exploit its approximation results to prove the reliability of the error indicator. We design and implement the corresponding adaptive polygonal anisotropic algorithm. Several numerical tests assess the superiority of the proposed algorithm in comparison with standard polygonal isotropic mesh refinement schemes.