Anisotropic Error Estimates of The Linear Virtual Element Method on Polygonal Meshes

TL;DR

This paper develops an advanced error analysis for the linear Virtual Element Method on polygonal meshes, including anisotropic elements, introducing new geometric assumptions and stabilization techniques.

Contribution

It introduces new geometric assumptions, a universal error equation, and a stabilization method for anisotropic polygonal meshes in VEM analysis.

Findings

Error estimates depend on shape regularity and anisotropy.

New stabilization improves error analysis for short edges.

Analysis extends to high aspect ratio elements.

Abstract

A refined a priori error analysis of the lowest order (linear) Virtual Element Method (VEM) is developed for approximating a model two dimensional Poisson problem. A set of new geometric assumptions is proposed on shape regularity of polygonal meshes. A new universal error equation for the lowest order (linear) VEM is derived for any choice of stabilization, and a new stabilization using broken half-seminorm is introduced to incorporate short edges naturally into the a priori error analysis on isotropic elements. The error analysis is then extended to a special class of anisotropic elements with high aspect ratio originating from a body-fitted mesh generator, which uses straight lines to cut a shape regular background mesh. Lastly, some commonly used tools for triangular elements are revisited for polygonal elements to give an in-depth view of these estimates' dependence on shapes.