## 1 Introduction  
Magnetotelluric (MT) sounding is a crucial technique widely employed in applied geophysics, e.g., for mineral resource surveys [1, 2]. The commonly used numerical modeling techniques for MT modeling include the finite difference method (FDM) [3], the finite-element method (FEM) [4], and the integral equation method [5]. On the other hand, apart from deriving governing equations directly for either the electric field or the magnetic field, another popular approach involves decomposing the Helmholtz equation into the electric scalar potential \(\phi\) and the magnetic vector potential \(\mathbf{A}\) under the Coulomb-gauged condition \(\
abla\cdot\mathbf{A}=0\)[6, 7]. This approach is also known as the \(\mathbf{A}-\phi\) method [8].  
Conductivity and magnetic permeability are two crucial physical parameters in MT forward modeling and inversion, and the phenomena of conductive anisotropy and non-zero susceptibility are commonly observed on the Earth [9, 10]. Ignoring the influences of conductive anisotropy or magnetic susceptibility may result in misinterpretations of MT data [11; 12]. Extensive research has been conducted on the forward modeling of MT in conductive anisotropic media [13; 14]. Notably, [PERSON] [15] presented a node-based FEM for MT modeling in three-dimensional (3D) generalized anisotropic media, while the nodal FE method for MT modeling lacks accuracy due to the limitations regarding simulating electric fields at electrical interfaces. [PERSON] and [PERSON] [16] investigated the spatial and frequency-dependent behavior of phase tensors and tipper vectors above a simple 3D anisotropic conductive anomaly using COMSOL MultiphysicsTM 4.3. [PERSON] et al. [17] proposed an algorithm utilizing an edge-based FEM to model 3D MT responses in arbitrary conductive anisotropic media, leading to insightful findings from their analysis of typical models. [PERSON] et al. [14] conducted qualitative and quantitative analyses on 3D synthetic models using the staggered-grid FDM. [PERSON] et al. [18] developed a 3D adaptive FEM that facilitates the simulation of irregular anomalies and topographies. Furthermore, [PERSON] et al. [19] introduced an FEM incorporating a divergence scheme specifically designed for axial conductive anisotropic media.  
However, research concerning forward modeling in 3D conductive anisotropic and magnetic media remains limited, with only a few studies addressing this topic, to the best of our knowledge. [PERSON] et al. [20] proposed a method that directly operates on the electric field, while [PERSON] et al. [9] presented an algorithm utilizing potentials through nodal FEM. In comparison to the results for the nodal FEM, the mixed nodal and edge-based FEM approach generates fewer unknowns, thereby enhancing computational efficiency. Therefore, we developed an alternative algorithm for MT modeling. In this study, we formulated the forward modeling of 3D MT in conductive anisotropic and magnetic media using **A** and \(\phi\) potentials employing the mixed nodal and edge-based FEM approach. This novel method offers distinct advantages for investigating the effects of conductive anisotropic and magnetic anomalies.  
The paper is structured as follows: First, we present a comprehensive description of the mixed nodal and edge-based element method for 3D MT modeling based on Coulomb-gauged potentials. Next, we compare the numerical solution obtained from the algorithm with both the analytical solution of a 1D anisotropic model and the results derived from a 3D generalized anisotropic model to validate its accuracy. Finally, we conduct in-depth investigations on several representative models, exploring the impact of conductivity in different directions on the horizontal electric field and examining how magnetic permeability influences regions abundant in magnetite.