Large-scale three-dimensional magnetotelluric forward modeling in anisotropic media using an extrapolation multigrid algorithm on non-uniform grids

In solving large-scale linear equations arising from three-dimensional (3-D) magnetotelluric (MT) forward modeling, parallelization techniques and multigrid methods are two of the widely-used acceleration approaches. Traditional geometric multigrid methods (GMG) depend on nested orthogonal grids and exhibit disadvantages when solving problems with jumping coefficients. For anisotropic problems, some special strategies like semi-coarsening and line/face smoothing are required, which limits the applications of such methods. Therefore, in this paper, an extrapolation cascadic multi-grid method (EXCMG) based on non-uniform hexahedral grids is proposed, and applied to accelerate the solution of large complex linear system sarising from 3-D MT anisotropic finite-element (FE) forward modeling. First, for the vector Helmholtz equation using Coulombgauged electromagnetic potentials a large sparse complex linear system can be derived with Galerkin weighted residual method. Then, a series of nested grids is generated through coarsening a fine nonuniform orthogonal grid level by level. Next, Richardson extrapolation and Lagrange's quadratic interpolation are employed to construct a new prolongation operator for nonuniform orthogonal grids. In the implementation of EXCMG, the highly accurate approximation to the FE solution on the next finer grid is constructed with the numerical solutions of the two coarse grids, and used as good initial guess for the multigrid smoother: Stable bi-conjugate gradient method (BiCGStab) with incomplete LU (ILU) preconditioner, to accelerate its convergence. Finally, the accuracy and efficiency of the algorithm are validated through several typical geoelectric models. Numerical experiments demonstrate that the proposed algorithm is appropriate for a wide frequency range, and the acceleration effect is obvious. Compared with preconditioned BiCGStab, the solving efficiency can be improved by dozens of times. It has been validated that our algorithm is capable of handling problems with strong anisotropy, as well as large-scale problems with more than 0.1 billion unknowns. The advantage of this algorithm is more evident for larger scale problems, and it is promising to apply EXCMG to other geophysical problems.