Finite‐difference solution of the three‐dimensional electromagnetic problem using divergence‐free preconditioners

Abstract

The modelling of the marine electromagnetic (EM) problem requires fine gridding to account for the seafloor bathymetry and to model complicated targets. This makes the computational cost of the problem large by using conventional Finite‐Difference (FD) solvers. To circumvent these problems, we employ a volume Integral Equation (IE) approach to arrive at a preconditioning operator. Such an approach significantly reduces the condition number and the size of the stiffness matrix. The preconditioner is divergence free and is based on a magnetic field formulation. The cost of constructing the preconditioning operator is much less than the one used to construct the operator in a standard IE approach. Further a homogenization technique that allows grids to be non‐conformal to the inhomogeneity interfaces is used. The optimal grid technique is used to extend the boundaries of the simulation domain to infinity. Some numerical results are presented in order to illustrate the accuracy and the effectiveness of the method.