This manuscript develops edge-averaged virtual element (EAVE) methodologies to address convection-diffusion problems effectively in the convection-dominated regime. It introduces a variant of EAVE that ensures monotonicity (producing an $M$-matrix) on Voronoi polygonal meshes, provided their duals are Delaunay triangulations with acute angles. Furthermore, the study outlines a comprehensive framework for EAVE methodologies, introducing another variant that integrates with the stiffness matrix derived from the lowest-order virtual element method for the Poisson equation. Numerical experiments confirm the theoretical advantages of the monotonicity property and demonstrate an optimal convergence rate across various mesh configurations.

翻译：本文发展了边缘平均虚拟元（EAVE）方法，以有效处理对流主导区域中的对流扩散问题。文中提出一种EAVE变体，在Voronoi多边形网格上确保单调性（生成$M$-矩阵），前提是其对偶网格为具有锐角的Delaunay三角剖分。此外，研究概述了EAVE方法的完整框架，引入另一种变体，该变体与基于最低阶虚拟元方法求解泊松方程所得的刚度矩阵相整合。数值实验验证了单调性特性的理论优势，并展示了在多种网格配置下的最优收敛速度。