Maxwell特征值问题混合元离散系统的快速算法

Finite element method has already become one of the most effective methods in numerical solving scientific engineering problems, Maxwell eigenvalue problems has quite important application value and prospect in electromagnetic field.. We study the current eigenvalue solver and construct more efficient preconditioned eigenvalue solver- preconditioned subspace iteration for the mixed finite element system, which arising from the Maxwell eigenvalue problem. We analysis the optimal convergence of this method, test the efficiency and stability by several numerical experiments.. Compare the exist works, this work has three innovation, first, we construct preconditioned subspace iteration for the Maxwell eigenvalue problems. We solve the preconditioned system by the multi-grid method based on a distribution matrix, and more important is we need not solve the precondition system exactly, which can save much computational work. Second, the multi-grid based on the distribution matrix can solve the general Maxwell equation saddle point system. Last, under the condition of solving the preconditioned system inexactly, we will prove the optimal convergence of the method.. This work is of great both theoretical and practical significance and promote the current algorithms and theories in some extent.

有限元方法是数值求解科学计算和工程问题的最有效方法之一，Maxwell特征值问题在电磁场领域具有重要的实际应用价值和前景。. 本课题旨在针对Maxwell特征值问题的混合元离散系统，深入研究当前常用的特征值解法器，在此基础上构造出更高效的预条件特征值解法器——预条件子空间迭代方法，分析算法的最优收敛性，并通过数值实验测试算法的高效性和稳定性。. 与已有工作相比，我们工作有三大创新：第一，我们构造的预条件子空间迭代法，其预条件行为通过基于分布矩阵的多重网格方法实现，更为重要的是该预条件行为不需要精确实现,这将会大大节省计算量；第二，基于分布矩阵的多重网格法将适用一般Maxwell方程的鞍点系统；第三，在不精确实现其预条件行为的情况下，我们将证明预条件子空间迭代法的最优收敛性。.本课题的研究是一项具有重要应用价值和理论意义的工作，在一定程度上对现有的算法和理论有所发展。

Maxwell 特征值问题一直都是科学和计算领域的重要研究课题。本项目针对Maxwell 特征值问题的混合元离散系统设计了相应的快速算法，理论上分析了快速算法的最优收敛性，编写了相应的程序，从数值算例上验证了算法的高效性。由于其离散系统是一个广义特征值问题，我们设计了反子空间迭代法，可以同时求出一批特征值。由于在反子空间迭代法中，需要求解相应的Maxwell方程的离散系统，我们又对其设计了相应的预条件子并结合gmres迭代法可以达到最优的效果。该工作可以说在一定程度上是对于Maxwell特征值问题在算法和理论上的进一步完善。