Effective polygonal mesh generation and refinement for VEM

Abstract

In the present work we introduce a novel refinement algorithm for two-dimensional elliptic partial differential equations discretized with Virtual Element Method (VEM). The algorithm improves the numerical solution accuracy and the mesh quality through a controlled refinement strategy applied to the generic polygonal elements of the domain tessellation. The numerical results show that the outlined strategy proves to be versatile and possibly applicable to each two-dimensional problem where polygonal meshes offer advantages. In particular, we focus on the simulation of flow in fractured media, specifically using the Discrete Fracture Network (DFN) model. A residual a-posteriori error estimator tailored for the DFN case is employed. We chose this particular application to emphasize the effectiveness of the algorithm in handling complex geometries. All the numerical tests demonstrate optimal convergence rates for all the tested VEM orders.