Fast Geometric Multigrid Preconditioned Finite- Difference Frequency-Domain Method for Borehole Electromagnetic Sensing

To model the responses of borehole electromagnetic sensing in complicated geological environments, the geometric multigrid preconditioned finite-difference frequency-domain (FDFD) method is examined on its numerical performance as a fast forward modeling solver in this work. The geometric multigrid method is employed as a preconditioner for the biconjugate gradient stabilized method to efficiently solve the linear system resulting from the finite-difference method based on the staggered Yee’s cell. The Gauss–Seidel method is utilized as both presmoother and postsmoother for the multigrid method, and the bilinear approach is employed to construct the restriction and prolongation operators between coarse and fine grids. The F-cycle coarsening scheme is used to balance the convergence and computational efficiency of the multigrid method. Numerical examples of borehole electromagnetic sensing are included to examine the performance of the method, which not only demonstrates its accuracy but also shows its capability as a fast solver for practical application. The geometric multigrid preconditioned FDFD method can work as a good candidate for 3-D modeling-based interpretation and evaluation for borehole electromagnetic sensing.