We present a virtual element method (VEM) for the numerical approximation of the electromagnetics subsystem of the resistive magnetohydrodynamics (MHD) model in two spatial dimensions. The major advantages of the virtual element method include great flexibility of polygonal meshes and automatic divergence-free constraint on the magnetic flux field. In this work, we rigorously prove the well-posedness of the method and the solenoidal nature of the discrete magnetic flux field. We also derive stability energy estimates. The design of the method includes three choices for the construction of the nodal mass matrix and criteria to more alternatives. This approach is novel in the VEM literature and allows us to preserve a commuting diagram property. We present a set of numerical experiments that independently validate theoretical results. The numerical experiments include the convergence rate study, energy estimates and verification of the divergence-free condition on the magnetic flux field. All these numerical experiments have been performed on triangular, perturbed quadrilateral and Voronoi meshes. Finally, we demonstrate the development of the VEM method on a numerical model for Hartmann flows as well as in the case of magnetic reconnection.

我们提出了一种虚拟元素方法（VEM），用于在两个空间维度上对电阻磁流体动力学（MHD）模型的电磁子系统进行数值逼近。虚拟元素方法的主要优点包括多边形网格的灵活性强以及对磁通量场的自动无散度约束。在这项工作中，我们严格证明了该方法的适定性和离散磁通量场的螺线管性质。我们还导出了稳定能估计值。该方法的设计包括用于节点质量矩阵构建的三种选择以及更多替代方案的标准。这种方法在VEM文献中是新颖的，可让我们保留通勤图财产。我们提出了一组数值实验，可以独立地验证理论结果。数值实验包括收敛速度研究，能量估计和磁通量场无散度条件的验证。所有这些数值实验都是在三角形，摄动的四边形网格和Voronoi网格上进行的。最后，我们在基于Hartmann流动的数值模型以及磁性重新连接的情况下，证明了VEM方法的发展。