Anisotropic mesh quality measures and adaptation for polygonal meshes

TL;DR

This paper develops three new anisotropic mesh quality measures for polygonal meshes and uses them to create an adaptive mesh method that improves the numerical solution of elliptic equations.

Contribution

It introduces three novel anisotropic mesh quality measures for polygonal meshes and applies them to develop an effective adaptive mesh method.

Findings

All three measures accurately assess mesh quality.

The adaptive method improves solution accuracy for elliptic equations.

Numerical tests confirm the effectiveness of the proposed approach.

Abstract

Anisotropic mesh quality measures and anisotropic mesh adaptation are studied for polygonal meshes. Three sets of alignment and equidistribution measures are developed, one based on least squares fitting, one based on generalized barycentric mapping, and the other based on singular value decomposition of edge matrices. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given metrics. Based on one of these sets of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.