We present an edge-based finite element modeling algorithm with a divergence correction for calculating controlled-source electromagnetic (CSEM) responses of a 3D conductivity earth model. We solve a curl-curl equation to directly calculate the secondary electric field in order to eliminate the source singularity. The choice of the edge-based finite element method enables us to properly handle the discontinuity of the normal component of electric fields across conductivity boundaries. Although we can solve the resulting complex-symmetric linear system of equations efficiently by a quasi-minimal residual 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 converge 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 correction technique with the secondary field solution using the quasi-minimal residual solver. We have found that applying the divergence correction intermittently during the iterative solution process ensures the calculation 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 the proposed algorithm with 1D and 3D models, and 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.
Minimal residual algorithm and matrix-vector information
