3-D MT Forward Modeling Using Finite-Element Method Based on Unstructured Geometric Multigrid

Abstract

In magnetotelluric (MT) forward modeling, the double-curl operator is associated with a large null space. In addition, the strong electrical contrast between the air and subsurface media leads to a large condition number for the equation system, causing traditional Krylov subspace methods to suffer from slow convergence or even divergence, especially at low frequencies. The existing geometric multigrid (GMG) methods based on structured hexahedral meshes offer high computational efficiency, yet their discretization with regular grids is difficult to adapt to complex geological structures, limiting their ability to model practical Earth structures. On the contrary, unstructured grids exhibit strong adaptability and allow for a more accurate representation of complex geological boundaries. By combining unstructured meshes with a GMG, it is possible to improve the efficiency of iterative solutions while maintaining good adaptability to complex structures. To this end, we propose a finite-element (FE) forward modeling method based on an unstructured GMG framework. The method constructs nested unstructured meshes progressively from coarse to fine, establishes restriction and prolongation operators between grids through basis function interpolation and integration, employs a V-cycle strategy to solve the linear system, and finally achieves 3-D MT forward modeling efficiently. The accuracy of the proposed algorithm is checked against the analytical solution for a homogeneous half-space model. Subsequently, a double-block model, a Dublin Test Model 1 (DTM-1), and a subduction-zone mantle plume model are constructed to validate the effectiveness of the proposed algorithm. The results demonstrate that the proposed GMG method outperforms traditional Krylov subspace methods (BICGSTAB-ILU, BICGSTAB-SOR, GMRES-ILU, and GMRES-SOR) in terms of iterations, computational time, and stability. It is more suitable for large-scale 3-D MT forward modeling under complex geological conditions.