频率域电磁法三维有限元正演线性方程组迭代算法

在三维频率域电磁法的正演模拟方法中，有限元方法具有计算精度高、适应性强的优点，近年来来得到了越来越多的关注.在正演过程中，主要的计算量集中在求解由偏微分方程组离散得到的线性方程组上，因此求解线性方程组关系着正演计算速度以及模拟精度.由于由有限元方法离散得到的复系数线性方程组条件数非常大，使用常规的迭代法和预条件很难收敛.目前大多数的研究工作采用直接解法，需要大量的计算机内存，限制了可求解问题的规模.本文研究了线性方程组的迭代解法，通过将复系数线性方程组转化为其实对称形式，构造分块对角预条件.在应用预条件的过程中，需要求解两个较小的实数方程，通过辅助空间解法求解.本文的算法适用于可控源电磁法和大地电磁法，对一系列的数值算例的模拟结果证明了迭代算法的效率，结果表明迭代算法可以在小于20次迭代内收敛，同时迭代次数与模型电阻率、问题规模和频率无关.

Research on the iterative solver of linear equations in three-dimensional finite element forward modeling for frequency domain electromagnetic method

In the forward modeling method for the three-dimensional frequency domain electromagnetic method, the finite element method has the advantages of high accuracy and strong adaptability and has received more and more attention in recent years. In the forward modeling process, the main computational effort is concentrated on solving the linear equations obtained by discretizing the partial differential equations. Therefore, the calculation speed and accuracy are determined by solving the linear equations. Since the condition numbers of the complex coefficient linear equations obtained by the finite element method are very large, it is difficult to converge using conventional iterative methods and preconditioners. Most of the current research work uses direct solvers, which require a large amount of computer memory, limiting the scale of the problem to be solved. In this work, the iterative methods of linear equations are studied. By rewriting the complex coefficient linear equations into its equivalent real form, the block diagonal preconditions are constructed. In the process of applying block-diagonal preconditioner, two smaller real number equations need to be solved and solved by the Auxiliary space Maxwell Solver. The algorithm in this work is applicable to both CSEM and MT problems. The simulation results of a series of numerical examples demonstrate the efficiency of the iterative algorithm. The results show that the iterative algorithm can converge in less than 20 iterations, and the number of iterations is independent of the model resistivity, the size of the problem and frequency.