一种基于亥姆霍兹分解的大地电磁测深有限元正演预条件解法

本研究针对大地电磁测深法有限元数值模拟中，迭代法求解线性方程组效率较低的问题，利用亥姆霍兹分解原理，将电场矢量双旋度方程的预条件问题转化为基于矢量位的泊松问题和基于标量位的拉普拉斯问题，并在四面体非结构化棱边元离散的情况下，借助节点元辅助网格离散上述预条件问题，进一步利用代数多重网格方法（AMG）实施求解，最终实现预条件算法.利用经典的COMMEMI理论模型进行试算并与前人的积分方程解进行对比，验证了本文数值模拟程序与预条件方法的正确性和可靠性.此外，利用不同自由度规模的实验模型对这一预条件算法的效率进行了测试.结果表明，这一算法可以有效地提升大地电磁测深法棱边有限元数值模拟迭代法的收敛性，计算效率较通用的不完全LU分解预条件算法明显更高；在较大自由度网格（>1000万）数值模拟计算中，其算法效率及内存占用相对直接解法有较大优势，也使小型工作站上利用较大自由度的有限元网格进行大地电磁测深数值模拟计算成为可能.

A Helmholtz decomposition based pre-condition method for magnetotelluric finite element numerical simulation

To improve the efficiency of iterative methods solving linear equations in finite element simulation of magnetotelluric sounding, we present a pre-conditioning method exploiting the Helmholtz decomposition of vector fields. Helmholtz decomposition is used to transform the pre-conditioned problem of vector curl-curl equation of time-harmonic Maxwell equation into a Poisson problem based on the vector potential and a Laplacian problem with the scalar potential. With tetrahedral unstructured edge element, the preconditioned problem is discretized on auxiliary node element grid and then solved by algebraic multigrid method (AMG). Then, the COMMEMI synthetic model is used to calculate responses, which is compared with those of integral equation method, and verifies the reliability of the simulation program and the pre-condition method. Additionally, the efficiency of the pre-condition algorithm is tested by discretizing the model with a large range of degrees of freedoms (DoFs). The results show that the new algorithm can significantly improve the convergence of iterative methods that applied to magnetotelluric numerical simulation and presents significantly higher efficiency than the common pre-conditioner with incomplete LU decomposition. Under the circumstances of comparably large DoFs (over 10 million), it manifests apparent advantages over direct solver in efficiency and memory consuming, and also makes the magnetotelluric numerical simulation with a finite element mesh of relatively large size possible on a mobile workstation.