An efficient extrapolation multigrid method for three-dimensional magnetotelluric finite element modeling using hierarchical semi-structured tetrahedron grids

In this study, we developed a novel three-dimensional (3D) magnetotelluric (MT) forward modeling algorithm based on the finite-element method and hierarchical semi-structured tetrahedron grids, enabling us to handle arbitrary complex models with irregular geometries and mountainous topography effectively. Firstly, we discretize the governing equation of the vector and scalar potentials by employing edge element and nodal element basis functions. This choice can reduce the condition number of the linear system and enhance the convergence of iterative solvers. Then, the extrapolation cascadic multigrid method (EXCMG) is utilized to solve the assembled linear system of equations. The EXCMG method incorporates extrapolation and high-order interpolation techniques on a tetrahedral grid. This approach is employed to construct a new multigrid prolongation operator, which allows for the generation of high-order approximations to the finite element solution on refined grids. To accelerate the convergence of the iterative solver, we utilize the numerical solutions obtained on two coarse meshes as the initial values for the multigrid smoothing operator, specifically SSOR-BiCGStab. Finally, we test the performance of our newly developed 3D MT forward modeling method by utilizing two synthetic models: COMMEMI-3D1 and the ridge model. The numerical computations demonstrate that our solutions closely match the published results. Additionally, we observe a significant reduction in solution time when employing our EXCMG method compared to the SSOR-BiCGStab and ILU-BiCGStab methods.