Finite-element EM modelling on hexahedral grids with an FD solver as a pre-conditioner

The finite-element (FE) method is one of the most powerful numerical techniques for modelling 3-D electromagnetic fields. At the same time, there still exists the problem of efficient and economical solution of the respective system of FE equations in the frequency domain. In this paper, we concentrate on modelling with adapted hexahedral or logically rectangular grids. These grids are easy to generate, yet they are flexible enough to incorporate real topography and seismic horizons. The goal of this work is to show how a finite-difference (FD) solver can be used as a pre-conditioner for hexahedral FE modelling. Applying the lowest order Nedelec elements, we present a novel pre-conditioned iterative solver for the arising system of linear equations that combines an FD solver and simple smoothing procedure. The particular FD solver that we use relics on the implicit factorization of the horizontally layered earth matrix. We assessed runtime and accuracy of the presented approach on synthetic and real resistivity models (topography of the Black Sea continental slope). We further compared performance of our program versus publicly available Mare2DEM, ModEM and MUMPS programs/libraries. Our examples involve plane-wave and controlled source modelling. The numerical examples demonstrate that the presented approach is fast and robust for models with moderate contrast, supports highly deformed cells, and is quite memory-economical.