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This study proposes a new algorithm for a higher-order vector finite element method based on two new types of second-order edge elements to solve the electromagnetic field diffusion problem in a 3D anisotropic medium. To avoid source singularity in the quasistatic variant of Maxwell’s function, a secondary field formulation was adopted. The modeling domain was discretized using two types of quadratic edge hexahedral elements, which were obtained using the edge unification method to reduce variables on each side of two conventional quadratic edge elements. Compared with the traditional quadratic element, the number of unknowns that needed to be solved was significantly reduced. The sparse linear equation of the finite element system was solved using an open-source direct solver called MUMPS. The numerical results demonstrated that the proposed algorithm has the same level of accuracy as the conventional vector finite element method and has a significant advantage over it in terms of computational cost.

Surface nuclear magnetic resonance (SNMR) is a geophysical method that can be used directly for detecting groundwater resources, and it has attracted the attention of many scholars. In this paper, we propose a new effective algorithm for numerical modeling of 3D SNMR data for arbitrary topography in a conductive medium. We adopt a total-field algorithm for solving the quasi-static variant of Maxwell’s equation and handle a complex-shaped loop source by discretizing the transmitter into electric dipoles, which can be further easily discretized into electric dipoles along the three directions of the Cartesian coordinate system. To solve the 3D SNMR forward-modeling problem quickly and accurately, a new element-integration system based on a new symmetric orthogonal rule is used for calculating the sensitivity (i.e., kernel) functions of all elements. The new rule is based on a special arrangement involving a cubic close-packed lattice structure and is characterized by fast convergence, positive weight, and symmetry. We apply the developed numerical algorithm to SNMR tomography of several typical hydrogeological models. The synthetic results show that higher precision can be achieved with few grids and nodes without increasing the computation time by using the new integration algorithm. In addition, we find that the topography and conductivity can affect the SNMR response, which needs to be considered while interpreting SNMR data.