Edge-Averaged Virtual Element Methods for Convection-Diffusion and Convection-Dominated Problems

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Shuhao Cao, Long Chen, Seulip Lee

Published: 2025, Last Modified: 20 Nov 2025J. Sci. Comput. 2025EveryoneRevisionsBibTeXCC BY-SA 4.0
Abstract: This paper 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.