###### Abstract  
Stability and convergence analyses for the domain decomposition finite element/finite difference (FE/FD) method are presented. The analyses are developed for a semi-discrete finite element scheme for time-dependent [PERSON]'s equations. The explicit finite element schemes in different settings of the spatial domain are constructed and a domain decomposition algorithm is formulated. Several numerical examples validate convergence rates obtained in the theoretical studies. finite element method; domain decomposition; finite difference method; stability analysis; convergence analysis; [PERSON]'s equations 1  
M: 1 2022  
(DDM) is constructed such that _FEM_ and _FDM_ coincide on the common, structured, overlapping layer between the two subdomains. The resulting domain decomposition approach at the overlapping layers can be viewed as a FE scheme which avoids instabilities at the interfaces. We decompose the computational domain such that _FEM_ and _FDM_ are used in different subdomains: _FDM_ in simple geometry and _FEM_ in the subdomain where more detailed information is needed about the structure of this subdomain. This allows the application of adaptive _FEM_ in such subdomain, see, e.g., [3; 9], and the references therein. Reliability and convergence of the domain decomposition method studied in this work are evident for the solution of coefficient inverse problems (CIPs) in \(\mathbb{R}^{3}\), see, e.g., [9], and the references therein. For the case of CIPs, the computational domain is split into subdomains such that a simple discretization scheme can be used in a large region and more refined discretization scheme is applied in smaller, but more critical, parts of the domain. In most algorithms for solution of electromagnetic CIPs, to determine the dielectric permittivity function inside a computational domain, a qualitative collection of experimental measurements at the boundary or in a neighborhood of it is necessary. In such cases, it is convenient to consider the numerical solution for [PERSON]'s equations in different subdomains with a constant dielectric permittivity function in some, adequately chosen ones, and non-constant in other ones. For time-dependent [PERSON]'s equations, the DD scheme of [4], analyzed in the present work, is used for the solution of different CIPs in order to determine the dielectric permittivity function in non-conductive media using simulated experimental data. There are many other studies relevant to this work, e.g., in [10; 11; 12; 13; 14; 15] and references therein. However, our analysis of DDM significantly differs from these works. In addition to optimal convergence of the constructed scheme, the novelty and scientific impact of the paper is in tackling coercivity (or lack of coercivity). More specifically, a challenging issue here is the lack of coercivity for the bilinear form in the presence of the time-derivative term. To circumvent this, a common approach is to split the bilinear form, separating the time-derivative term, and deriving the coercivity for the remaining sub-bilinear form. Then one may derive the convergence of the entire scheme with a time-derivative term. An outline of this paper is as follows. In Sections 2 and 3 we introduce the mathematical model, and briefly present the structure of domain decomposition FE/FD schemes and communication between the two schemes. In Section 4, we describe the domain decomposition FE/FD method for the numerical solution of [PERSON]'s equations and set up adequate finite element and finite difference schemes. Section 5 is devoted to the stability analysis for the original bilinear form, (with a norm involving the time derivative term). In Section 6, we derive optimal a priori error estimates in finite element approximation for the semi-discrete (spatial discretization) problem on the entire domain, which is easily reduced to the FE subdomain, having known boundary data (the restriction of FD solution to the boundary of the FE subdomain).