[Paper Review] Benchmark of Polygon Quality Metrics for Polytopal Element Methods

Marco Attene, Silvia Biasotti+7 more|arXiv (Cornell University)|May 27, 2019

Advanced Numerical Methods in Computational Mathematics20 references4 citations

TL;DR

This paper introduces a systematic benchmark to evaluate the correlation between polygonal mesh quality metrics and the performance of Polytopal Element Methods (PEM), focusing on the Virtual Element Method (VEM). By testing a wide range of polygon shapes and metrics—including area, angles, edge lengths, and kernel properties—it identifies weaker shape-regularity criteria under which PEM solvers remain stable and accurate, revealing that current theoretical constraints are overly restrictive and that VEM performs reliably even on highly irregular polygons.

ABSTRACT

Polytopal Element Methods (PEM) allow to solve differential equations on general polygonal and polyhedral grids, potentially offering great flexibility to mesh generation algorithms. Differently from classical finite element methods, where the relation between the geometric properties of the mesh and the performances of the solver are well known, the characterization of a good polytopal element is still subject to ongoing research. Current shape regularity criteria are quite restrictive, and greatly limit the set of valid meshes. Nevertheless, numerical experiments revealed that PEM solvers can perform well on meshes that are far outside the strict boundaries imposed by the current theory, suggesting that the real capabilities of these methods are much higher. In this work, we propose a benchmark to study the correlation between general 2D polygonal meshes and PEM solvers. The benchmark aims to explore the space of 2D polygonal meshes and polygonal quality metrics, in order to identify weaker shape-regularity criteria under which the considered methods can reliably work. The proposed tool is quite general, and can be potentially used to study any PEM solver. Besides discussing the basics of the benchmark, in the second part of the paper we demonstrate its application on a representative member of the PEM family, namely the Virtual Element Method, also discussing our findings.

Motivation & Objective

• To close the gap between theoretical shape-regularity constraints and practical solver performance in Polytopal Element Methods (PEM).
• To systematically evaluate how various geometric properties of 2D polygonal elements affect the accuracy and conditioning of PEM solvers.
• To identify permissive quality metrics that still ensure reliable performance, enabling more flexible mesh generation.
• To provide a modular, reusable benchmark framework applicable to any PEM solver, including the Virtual Element Method (VEM).

Proposed method

• The benchmark evaluates 12 polygon quality metrics per element, aggregated as min, max, and average to form 36 mesh-level metrics.
• A parametric family of polygons is generated to progressively stress individual metrics, combined with random polygons to avoid bias.
• The Virtual Element Method (VEM) is used as a representative PEM solver, with solution accuracy and linear system conditioning as performance indicators.
• The relative error in the L∞ and L2 norms is computed using vertex degrees of freedom, approximating the true error via the ∇-projection operator.
• A modular framework is designed to allow extension to other PEM methods and additional quality metrics.
• The study uses a model Poisson problem on polygonal domains to test convergence and stability under varying mesh quality.

Experimental results

Research questions

• RQ1 Which polygon quality metrics most strongly correlate with the accuracy and conditioning of PEM solvers, particularly the Virtual Element Method?
• RQ2 To what extent can shape-regularity constraints be relaxed without degrading solver performance in polytopal methods?
• RQ3 How do extreme polygonal shapes—such as non-star-shaped or highly distorted elements—impact the convergence and stability of PEM solvers?
• RQ4 Can geometric features like inscribed/circumscribed circles, kernel size, and edge length ratios be used to predict solver reliability?
• RQ5 What are the practical limits of mesh quality beyond which PEM performance breaks down?

Key findings

• The Virtual Element Method maintains optimal convergence rates and good conditioning even on highly irregular polygons that violate standard star-shapedness and shape-regularity assumptions.
• Metrics such as the ratio of the inradius to diameter and the minimum angle show strong correlation with solver performance, suggesting their use in mesh quality assessment.
• Non-star-shaped elements with complex geometries, such as snowflake-like shapes, still yield accurate solutions, indicating that current theoretical constraints are overly conservative.
• The benchmark reveals that the combination of small inradius and large diameter is a key predictor of poor conditioning, while edge length uniformity and vertex spread contribute to stability.
• The study identifies that kernel size and circumradius-to-inradius ratio are among the most predictive metrics for solver reliability.
• The framework successfully detects performance degradation before it becomes critical, enabling early identification of problematic mesh elements.