Numerical Solution of the Three-Dimensional Time-Harmonic Maxwell Equations by DG Method Coupled with an Integral Representation (original) (raw)

This work is dedicated to the numerical results and the implementation of the method coupling a discontinuous Galerkin with an integral representation (CDGIR). The originality of this work lies in the choice of discretization by discontinuous Galerkin element and a mixed form for Maxwell's equations. The numerical tests justify the effectiveness of the proposed approach. Keywords: Finite element method, Maxwell equations, Discontinuous Galerkin method, fictitious domain, integral representation, time-harmonic. I. INTRODUCTION Mathematically, the phenomenon of the electromagnetic waves propagation is generally modeled by the system of equations known as the Maxwell equations. There are two modes of the Maxwell equations to be treated, a first mode that is known by the time domain Maxwell equations in which the evolution of electromagnetic fields is studied as a function of time and the second mode that is known by the frequency domain Maxwell equations where one studies the behavior of electromagnetic fields when the source term follows a harmonic dependence in time. Numerical modeling has become the most important and widely used tool in various fields such as scientific research. The finite-difference methods (FDM), the finite element methods (FEM) and the finite volume methods (FVM) are the three classes of methods known for the numerical resolution of the problems of electromagnetic waves propagation. In 1966, Yee cited the first efficient method in [42] which is the finite-difference methods in the time domain (FDMTD). When diffraction problems are posed in unbounded domain, the use of these methods induces a problem. In order to solve it, two techniques are used. The first consists in reducing to a bounded domain by truncating the computational domain, then it is necessary to impose an artificial condition on the boundary on the truncation boundary. The second technique consists in writing an equivalent problem posed on the boundary of the obstacle, it is therefore what is called the theory of integral equations. The numerical resolution can then be done by discretizing the problem by collocation (method of moments, method of singularity) or by a finite element discretization of the boundary. In 1980, Nedelec introduces the edge finite element method developed in [31] which is also available in [29, 30]. With the conservation of energy, this method also possesses several advantages; it allows to treat unstructured meshes (complex geometries) as it can be used with high orders (see [41, 24, 29]). In recent years, research has revealed a new technique known as Discontinuous Galerkin Methods (GDM); this strategy is based on combining the advantages of FEM and FVM methods since it approaches the field in each cell by a local basis of functions by treating the discontinuity between neighboring cells by approximation FVM on the flows. Initially, these methods have been proposed to treat the scalar equation of neutron transport (see [35]). In the field of wave propagation, precisely for the resolution of the Maxwell equations in the time domain, many schemes are based on two forms of formulations: a concentrated flux formulation (see [16, 34]) and an upwind flux formulation (see [22, 12]). Discontinuous Galerkin methods have shown their effectiveness in studying the problem with discrete eigenvalues (see [23]). In frequency domain, for the resolution of Maxwell equations, the majority consider the second order formulation (see [25, 32, 33]), as others study the formulation of the first order as in [6, 20]. This strategy of the CDGIR method allows us to write a problem in an unbounded domain into an equivalent problem in a domain bounded by a fictitious boundary where a transparent condition is imposed. This transparent condition is based on the use of the integral form of the electric and magnetic fields using the Stratton-chu formulas (see [7]). This process has been studied, in the