外推瀑布式多网格法及其在三维地电磁场计算中的应用

三维地电磁场的高精度快速计算对提高勘探电磁法的资料解释水平具有重要的理论和实际意义。多网格法是快速求解大规模科学工程计算问题最有效的方法之一。本项目首次将外推瀑布式多网格法(EXCMG)应用到地电磁场正演计算中, 构造并实现基于EXCMG的三维地电磁场快速、高精度计算方法。采用局部加密结构化网格和分片均匀网格, 既方便对点电源或异常电位体附近进行局部加密, 又保留了有限元所固有的超收敛特性。重点研究有限元误差渐近展开及高精度外推公式, 提出基于局部加密网格的EXCMG算法。以此为基础, 研究并行EXCMG算法, 利用二次场和稀疏矩阵压缩存贮等技巧, 在并行机上实现上亿自由度的三维地电磁场的快速、高精度数值模拟。通过大量典型模型的计算与分析, 测试并改进算法,为后续三维电磁场的反演研究奠定基础。本项目的预期成果有望在计算地球物理领域得到广泛的应用，具有重要的理论意义和工程实用价值。

The implementation of fast and high precision computations of 3D geoelectromagnetic field has important theoretical and practical significance to improve the quality of data interpretation for electromagnetic prospecting methods. Multigrid method is one of the most efficient methods for solving large-scale scientific and engineering problems. This project, the first of its kind, will apply the extrapolation cascadic multi-grid method (EXCMG) to numerical simulations of electromagnetic fields, and propose and implement a fast and accurate calculation method for 3D geoelectromagnetic field simulations based on EXCMG. By using locally refined and structured grids (such as graded grids) and piecewise uniform grids, it is convenient to locally refine grids near the point source or abnormal electrical body, and the inherent superconvergence of FEM is retained. The project focuses on asymptotic error expansions and high-precision extrapolation formulas for FEM, and proposes an EXCMG algorithm based on locally refined grids. On this basis, a parallel implementation of the EXCMG algorithm will be developed. Combined with the secondary field and sparse matrix compression techniques, high-precision numerical simulation of 3D geoelectromagnetic field with hundreds of millions of degrees of freedom can be implemented rapidly on a machine with high parallel computation capability. Through a large number of numerical calculations and analysis of typical models, we will test and improve the algorithm, and lay the foundation of the subsequent study of 3D electromagnetic field inversion. The expected conclusion of this project will include algorithms and actual implementations that can be widely used in the field of computational geophysics, which will prove to have important theoretical and practical values.