Divergence-free solutions to electromagnetic forward and adjoint problems: a regularization approach

We discuss a regularization approach to improving numerical stability of the frequency domain Maxwell equations of electromagnetic geophysics. To enforce divergence-free conditions we add a scaled grad-div operator to the curl-curl equation for electric fields. This deflates the null space of the curl-curl operator and significantly reduces the condition number of the linear system derived for the finite difference (FD) approximation, resulting in faster and more stable iterative solution. We explicitly discuss extensions needed to solve the adjoint problems and consider two apparently different approaches to the sensitivity calculation, which we show are ultimately equivalent. To complete assessment of the new approach in practice we have implemented the modified solver in the ModEM inversion code, and compared it to more standard methods (based on solving with a divergence correction) on inversion of a real data set. This, and simpler tests of the forward solver in isolation, demonstrate that, with appropriate scaling of the added grad-div term, the regularization approach can dramatically improve the efficiency of Krylov subspace methods. Run times for the inversion test are reduced by almost a factor of three, making 3-D FD inversion significantly more practical.