To solve the multi-scale model of geoelectric field induced by geomagnetic storms, the domain decomposition method is put to use in this paper. As for each decomposed domain, modeling and solving can be worked out by means of finite element method (FEM). In order to simplify the iterative process and avoid the processing of the boundary conditions on the interface generated by the non-overlapping domain decomposition, the overlapping domain decomposition method is adopted. The potential function on the virtual boundary is chosen as the unknown variable, based on the linear relationship in computation of FEM, the problem of multi-scale induced geoelectric field is finally transformed into an asymmetric complex linear equation. In order to save computational memory, the GMRES iterative algorithm of domain decomposition is deduced without directly solving the specific coefficient matrix of the equations according to the mathematical principles of Arnoldi's orthogonalization algorithm and generalized minimal residue (GMRES) method. In this paper, a small scale model is derived from the whole model of the large scale induced geoelectric field, and a multi-scale solution model is formed. For 2D and 3D cases, the iterative variable selection and the derivation of the iterative process are described. Taken the results of FEM direct solution as benchmark, the correctness is verified. Besides, the efficiency of the proposed algorithm is verified by comparing with direct iteration and relaxation iteration in 2D case. The proposed algorithm is used to calculate a numerical example of multi-scale geoelectric field, which provides a new method for the calculation of multi-scale alternating electromagnetic field.%采用区域分解法求解多尺度磁暴感应地电场模型,对每个分解区域采用有限元法(FEM)建模求解.为简化迭代计算过程,避免非重叠区域分解在交界面上边界条件的处理,本文采用重叠型的区域分解方式.选择虚拟边界上的位函数作为未知变量,基于有限元计算的线性关系,将多尺度磁暴感应地电场问题最终转化为非对称复数线性方程组.为节省计算内存,在不直接具体求解方程组系数矩阵的情况下,依据Arnoldi正交化算法和广义极小残量(GMRES)法的数学原理,推导区域分解的 GMRES迭代算法.本文选择在整体大尺度磁暴感应地电场求解模型中,分解出小尺度模型,形成多尺度求解模型.针对二维情形和三维情形,分别阐述了迭代的变量选取和迭代过程.以有限元法直接求解的计算结果为基准进行对比,验证了算法的正确性.二维情况下,通过与直接迭代和松弛迭代的对比,验证了本文算法的高效性.利用本文的算法计算多尺度磁暴感应地电场的算例模型为多尺度交变电磁场计算提供了一种新方法.
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