## 2 The Mathematical Model  
The Cauchy problem for the electric field \(E(x,t)=(E_{1},E_{2},E_{3})(x,t)\), \(x\in\mathbb{R}^{3}\), \(t\in[0,T]\), of [PERSON]'s equations, under the assumptions that the dimensionless relative magnetic permeability of the medium is \(\mu_{r}\equiv 1\), and the electric volume charges are zero, is given by  
\[\frac{1}{c^{2}}\varepsilon_{r}(x)\frac{\partial^{2}E}{\partial t ^{2}}+\
abla\times\
abla\times E =-\mu_{0}\sigma(x)\frac{\partial E}{\partial t}, \tag{1}\] \[\
abla\cdot(\varepsilon E) =0,\]  
where \(\varepsilon_{r}(x)=\varepsilon(x)/\varepsilon_{0}\) and \(\sigma(x)\) are the dimensionless relative dielectric permittivity and electric conductivity functions, respectively. \(\varepsilon_{0}\) and \(\mu_{0}\) are the permittivity and permeability of the free space, respectively, and \(c=1/\sqrt{\varepsilon_{0}\mu_{0}}\) is the speed of light in free space. In this paper we consider problem (1) in non-conductive media, i.e., \(\sigma\equiv 0\), and hence, to begin with, we shall consider the following initial value problem \[\left\{ \begin{aligned} \varepsilon(x)\frac{\partial^{2}E}{ \partial t^{2}}+\
abla\times\
abla\times E&=0,& x\in\mathbb{R}^{3},\ t\in(0,T],\\ \
abla\cdot(\varepsilon E)&=0,&\text{ in }\ \Omega \times(0,T],\\ E(x,0)=f_{0}(x),&\frac{\partial E}{\partial t}(x,0) &=f_{1}(x),\ \ x\in\mathbb{R}^{3},\ t\in(0,T].\end{aligned} \right. \tag{2}\]  
To solve problem (2) numerically, we need to consider it in a bounded domain \(\Omega\subset\mathbb{R}^{n},\ n=2,3\), with boundary \(\partial\Omega\), and employ a split scheme on \(\Omega\): a hybrid, finite element/finite difference scheme, a kind of domain decomposition described in Algorithm 1. In this approach, we divide the computational domain \(\Omega\) into two subdomains, \(\Omega_{FEM}\) and \(\Omega_{FDM}\) such that \(\Omega=\Omega_{FEM}\cup\Omega_{FDM}\) (note that \(\Omega_{FEM}\cap\Omega_{FDM}\
eq\emptyset\)), and \(\Omega_{FEM}\) is a subset of the convex hull of \(\Omega_{FDM}\). The function \(\varepsilon(x)\) is assumed to be constant in \(\Omega_{FDM}\), and bounded and smooth in \(\Omega_{FEM}\). The communication between \(\Omega_{FEM}\) and \(\Omega_{FDM}\) is arranged using an overlapping mesh structure through a two-element thick layer around \(\Omega_{FEM}\), as shown by the blue and green common boundaries in Figure 1. The blue boundary is the boundary of \(\Omega_{FEM}\), the green boundary is the inner boundary of \(\Omega_{FDM}\), and the red boundary is the outer boundary of \(\Omega\) (which is also the boundary of the original domain \(\Omega\)). ```
1:On the mesh \(\Omega_{FDM}\), where \(FDM\) is used, update the Finite Difference (FD) solution. 2:On the mesh \(\Omega_{FEM}\), where \(FEM\) is used, update the finite element (FE) solution. 3:Copy the FE solution obtained at nodes \(\omega_{\circ}\) (nodes on the green boundary of Figure 1) as a boundary condition on the inner boundary for the FD solution in \(\Omega_{FDM}\). 4:Copy the FD solution obtained at nodes \(\omega_{\circ}\) (nodes on the blue boundary of Figure 1) as a boundary condition for the FE solution on \(\partial\Omega_{FEM}\) of \(\Omega_{FEM}\). 5:Apply boundary condition at \(\partial\Omega\) at the red boundary of \(\Omega_{FDM}\). ```  
**Algorithm 1** Domain decomposition process for the hybrid FE/FD scheme  
The key idea with such a decomposition is that it allows applying different numerical methods in different computational domains. For the numerical solution of (2) in \(\Omega_{FDM}\), we use the finite difference method on a structured mesh. In \(\Omega_{FEM}\), we use finite elements on a sequence of unstructured meshes \(K_{h}=\{K\}\), with elements \(K\) consisting of triangles in \(\mathbb{R}^{2}\) and tetrahedron's in \(\mathbb{R}^{3}\), both satisfying the minimal angle condition. This approach combines the flexibility of the finite element and the efficiency of the finite difference in terms of speed and memory usage and fits well for the reconstruction algorithm presented below. **Assumption 1**.