标题： 边元鞍点系统及其在时谐麦克斯韦方程中的预条件分析 显示英文标题

标题： Preconditioners and Their Analyses for Edge Element Saddle-point Systems Arising from Time-harmonic Maxwell Equations

摘要： 我们将提出并分析一些新的预条件器，用于解决三维时谐麦克斯韦方程组的边元离散化所产生的鞍点系统。 我们将首先考虑波数为零的鞍点系统，对于这种情况，我们展示了奇异旋度-旋度系统解与非奇异鞍点系统解之间的重要关系，然后证明了可以通过Hiptmair-Xu求解器高效求解鞍点系统。 对于波数不为零的鞍点系统，我们将展示当波数较小时，带有新预条件器的PCG可用于非奇异系统，而当波数较大时，如预条件MINRES等方法可用于一些现有和新的预条件器。 分析并比较了现有和新预条件器所产生预条件系统的谱特性，并通过数值实验演示和比较了这些预条件器的效率。 显示英文摘要

摘要： We shall propose and analyze some new preconditioners for the saddle-point systems arising from the edge element discretization of the time-harmonic Maxwell equations in three dimensions. We will first consider the saddle-point systems with vanishing wave number, for which we present an important relation between the solutions of the singular curl-curl system and the non-singular saddle-point system, then demonstrate that the saddle-point system can be efficiently solved by the Hiptmair-Xu solver. For the saddle-point systems with non-vanishing wave numbers, we will show that the PCG with a new preconditioner can apply for the non-singular system when wave numbers are small, while the methods like preconditioned MINRES may apply for some existing and new preconditioners when wave numbers are large. The spectral behaviors of the resulting preconditioned systems for the existing and new preconditioners are analyzed and compared, and numerical experiments are presented to demonstrate and compare the efficiencies of these preconditioners.