Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth: finite-element frequency-domain approach

Abstract

We formulate a 3-D finite-element frequency-domain (FEFD) solution for electromagnetic (EM) diffusion and present efficient solution strategies. A system of FEFD equations is pre-conditioned by incomplete LU (ILU) and subsequently solved by the quasi-minimal residual (QMR) method. A rule of thumb for choosing an effective drop tolerance of ILU is proposed. When multiple sources are simulated in a given model, ILU is computed only once and is reused as a pre-conditioner for multiple QMR computations with different source vectors. Resulting solution vectors are also bootstrapped to reduce the number of QMR iterations required for the convergence. We demonstrate that when conductivity structures of an earth model and source frequencies are updated/perturbed, ILU that is computed from the previous model is still an effective and useful pre-conditioner for new forward modelling problems. Using the reusability of ILU, we also propose a new efficient way to overcome the slow convergence rate of the iterative FEFD solution in the static limit. We show that the reuse of ILU and solution bootstrapping serve as effective strategies for improving the computational efficiency of the iterative FEFD solution. Finally, we apply the proposed efficient solution strategies to marine EM survey scenarios in complex offshore models and further demonstrate their effectiveness.