Three-dimensional controlled-source electromagnetic forward modeling by edge-based finite element with a divergence correction

We have developed an edge-based finite-element (FE) modeling algorithm with a divergence correction for calculating the controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field to eliminate the source singularity. The choice of the edge-based FE method en-ables us to properly handle the discontinuity of the normal com-ponent of electric fields across conductivity boundaries. Although we can solve the resulting complex-symmetric linear system of equations efficiently by a quasiminimal residual (QMR) method preconditioned with an incomplete Cholesky decomposition for the high-frequency band, the iterative solution process encounters a common problem in the field formulation and does not con -verge within a practically feasible number of iterations for low frequencies. To overcome this difficulty and to accelerate the iterative solution process in general, we combine a divergence cor-rection technique with the secondary field solution using the QMR solver. We have found that applying the divergence correction in-termittently during the iterative solution process ensures the cal-culation of sufficiently accurate electric and magnetic fields and can significantly speed up the solution process by more than an order of magnitude. We have tested the efficiency and accuracy of our algorithm with 1D and 3D models, and we have found that the divergence correction technique is able to guide the electric field to satisfy the boundary conditions across conductivity interfaces. Although there is a computational overhead required for applying the divergence correction, that cost is significantly offset by the substantial gains in the solution accuracy and speed-up. The work makes the field-based curl-curl formulation using edge elements an efficient and practical method for CSEM simulations.