磁流体动力学耦合系统的虚元法
The virtual element method for the coupled system of magneto-hydrodynamics
摘要：
在这项工作中，我们审查的框架虚拟单元法（VEM）的模型在磁流体动力学（MHD），它结合了电磁场和流体流动之间的耦合，并允许我们构建新的离散化模拟现实的现象在MHD。首先，我们研究两个链的空间近似的电磁和流体流动的模型组件。然后，我们表明，这种VEM近似将产生发散自由离散磁场，在任何模拟MHD的一个重要属性。我们提出了一个线性化的策略来解决的VEM近似，尊重发散自由条件的磁场。这种线性化将要求在每次非线性迭代中求解线性系。我们研究这些线性系，并表明，它们代表适定的鞍点问题。最后，我们提出了数值实验探索的VEM应用于子系统描述的电磁性能。第一组实验提供了关于该方法的收敛速度以及磁场上的无发散条件的证据。在第二组中，我们提出了一个模型，磁重联在一个网格，其中包括一系列的悬挂节点，我们用它来校准的方法的分辨率。磁场重联现象发生在磁畴中心附近，在此区域网格分辨率较细，可获得较高的分辨率。
英文摘要：
In this work, we review the framework of the Virtual Element Method (VEM) for a model in magneto-hydrodynamics (MHD), that incorporates a coupling between electromagnetics and fluid flow, and allows us to construct novel discretizations for simulating realistic phenomenon in MHD. First, we study two chains of spaces approximating the electromagnetic and fluid flow components of the model. Then, we show that this VEM approximation will yield divergence free discrete magnetic fields, an important property in any simulation in MHD. We present a linearization strategy to solve the VEM approximation which respects the divergence free condition on the magnetic field. This linearization will require that, at each non-linear iteration, a linear system be solved. We study these linear systems and show that they represent well-posed saddle point problems. We conclude by presenting numerical experiments exploring the performance of the VEM applied to the subsystem describing the electromagnetics. The first set of experiments provide evidence regarding the speed of convergence of the method as well as the divergence-free condition on the magnetic field. In the second set we present a model for magnetic reconnection in a mesh that includes a series of hanging nodes, which we use to calibrate the resolution of the method. The magnetic reconnection phenomenon happens near the center of the domain where the mesh resolution is finer and high resolution is achieved.
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