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 applicable to any 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.

翻译：本文提出了一种新颖的细化算法，用于处理采用虚拟单元法（VEM）离散化的二维椭圆型偏微分方程。该算法通过针对域剖分中通用多边形单元实施受控细化策略，提高了数值解的精度和网格质量。数值结果表明，所提出的策略具有通用性，可应用于任何从多边形网格中获益的二维问题。我们特别聚焦于裂隙介质中的流动模拟，具体采用离散裂隙网络（DFN）模型，并使用了针对DFN情况定制的后验残差误差估计器。选择这一特定应用是为了强调算法在处理复杂几何形状方面的有效性。所有数值测试均显示，在测试的所有VEM阶数下均达到了最优收敛率。