Fast iterative solver for 3D finite-element modeling of controlled-source electromagnetic responses with vector and scalar potentials

Abstract

The controlled-source electromagnetic (CSEM) forward modeling with vector and scalar potentials could provide insights into the inductive and galvanic effects. We have developed a finite-element (FE) forward modeling algorithm for the CSEM with vector and scalar potentials. The first-order vector and nodal shape functions are used for the vector and scalar potentials, respectively. To mitigate the null space and thus the nonuniqueness of the potentials caused by the curl operator, we introduce a new preconditioner constructed from an incomplete Cholesky decomposition with zero fill-ins of the Laplacian approximation to the resulted matrix after the FE discretization. We implemented a preconditioned quasiminimal residual method to iteratively solve the resulting linear system with the new preconditioner. We first verified the accuracy of the algorithm through comparison against the analytic solutions obtained for a three-layered earth model. The new iterative modeling algorithm converges much faster compared with the modeling algorithm with the preconditioner constructed from the original stiffness matrix. The efficiency and accuracy are further illustrated by comparison with the modeling scheme based on coupled potentials with explicit enforcement of the Coulomb gauge condition for the vector potential by modeling three 3D models. In addition, the accuracy of our algorithm is demonstrated via a comparison of our numerical solution for a 3D model with the numerical solution obtained from the integral equation method. We also present the inductive and galvanic components of the horizontal secondary electric field for numerical solutions obtained with direct and iterative solvers. The numerical test demonstrated that, aside from the efficiency improved with the new iterative solver, the unstable and nonuniqueness problem for the potentials is eliminated by the new preconditioner with the Laplacian approximation involved.