论文标题
平行块预处理，用于时间相关的麦克斯韦方程的虚拟元素离散器
Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations
论文作者
论文摘要
这项研究的重点是对平行块预处理的构建和数值验证，用于三维Maxwell方程的低阶虚拟元素离散化。虚拟元素方法（VEM）是针对部分微分方程（PDE）的数值近似的最新技术，它将有限元概括为多地计算网格。到目前为止，VEM已扩展到PDE所描述的几个问题，最近也扩展到了时间依赖的Maxwell方程。当隐式执行PDE的时间离散化时，在每个时间步长时间，必须解决大规模且条件不良的线性系统，如果麦克斯韦方程式在麦克斯韦方程式中，则尤其具有挑战性，因为H（div）和H（curl）离散空间都存在。我们在这里提出了一个并行的预处理，该预处理可利用线性系统矩阵的Schur补体块分解，并由H（div）块的Jacobi预处理组成，H（curl）块的辅助空间预处理器组成。已经完善了几个平行的数值测试，以研究求解器相对于网状细化，多面体元素的形状，时间步长和VEM稳定参数的鲁棒性。
The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been perfomed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.
