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.