Theoretical Analysis of Geo‐electromagnetic Modeling on Coulomb Gauged Potentials by Adaptive Finite Element Method

Abstract

AbstractThere are several serous problems in geo‐electromagnetic modeling by the direct solving method at present. Firstly the pseudo‐solutions physically exists in geo‐electromagnetism fields. Secondly, the discontinuous conditions of electromagnetic field at the conductivity interfaces have limited the application of the nodal‐based finite element method. In this paper, in terms of Coulomb gauged continuous magnetic vector potential and electric scalar potential and the newly generated adaptive finite‐element method, we propose a new fast and accurate method for forward modeling of geo‐electromagnetic field. Based on the typical electromagnetic boundary value problem, we introduce the general formula system of magnetic vector potential and electric scalar potential by Coulomb gauged potentials, and have proved the continuous properties of magnetic vector potential and scalar potential. By means of Galerkin weighted residual technique, we have deduced the corresponding weak integral form of electromagnetic vector and scalar potentials formulation and then in terms of unstructured Hierarchal tetrahedron finite‐element families, we have presented the final finite‐element linear equations. After this linear equation has been solved, we employed the super‐convergence recovery technique to estimate the element and global error indicators on the current mesh. In terms of our optimal adaptive mesh refinement or adaptive order updating strategies, we have analyzed the maximum time complexity and error convergence rate of our adaptive process. Theoretical analysis indicates we could adopt optimal computational cost to achieve much accurate numerical solutions of geo‐electromagnetic field by nearly exponential convergence rate of numerical errors. Therefore, this new technique can be applied as a theoretical guide in wide range of geophysical electromagnetic explorations.