Maxwell's equations are a system of partial differential equations that govern the laws of electromagnetic induction. We study a mimetic finite-difference (MFD) discretization of the equations which preserves important underlying physical properties. We show that, after mass-lumping and appropriate scaling, the MFD discretization is equivalent to a structure-preserving finite-element (FE) scheme. This allows for a transparent analysis of the MFD method using the FE framework, and provides an avenue for the construction of efficient and robust linear solvers for the discretized system. In particular, block preconditioners designed for FE formulations can be applied to the MFD system in a straightforward fashion. We present numerical tests which verify the accuracy of the MFD scheme and confirm the robustness of the preconditioners.
翻译:马克斯韦尔的方程式是适用于电磁感应法的局部差异方程式系统。我们研究了这些方程式的微微有限差异(MFD)分解(MFD),这保留了重要的内在物理特性。我们表明,在大规模排除和适当缩放后,MFD的分解相当于一个结构保留有限元素(FE)计划。这样就可以利用FE框架对MFD方法进行透明的分析,并为为分解系统建造高效和稳健的线性线性解决器提供一个途径。特别是,为FE配方设计的区块先决条件可以直截了当的方式适用于MFD系统。我们提出了数字测试,以核实MFD计划的准确性,并证实先决条件的健全性。