FEM APPROXIMATION OF MAXWELL EQUATIONS: THE SOURCE PROBLEM, EIGENPROBLEM, AND ELECTROMAGNETIC WAVES

2023-1-25
Dağlı, Tugay
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In this thesis, edge-based finite element method (FEM) approximations of Maxwell's equations describing the relationship between the space variables and sources along the electromagnetic field are considered. In particular, the lowest-order Nédélec basis functions are implemented to construct the FEM model of the Maxwell source problem, Maxwell eigenvalue problem (EVP), and electromagnetic wave propagation problem. A computational model is constructed to conduct all these problems in the same framework of an approximation formalism. The convergence properties of the Maxwell EVP formulation are analyzed by applying the spectral theory with those of the associated boundary value source problem. Therefore, the analyses for both of these problems are given together with the corresponding numerical results validating the theoretical features. Moreover, a comparison between the two convergent FEM approximations of the Maxwell EVP that utilizes the lowest-order Nédélec and the linear Lagrange basis functions is performed. This comparison is done by using a special triangulation, namely, a Powell-Sabin type, of the domain that contains a strong singularity. The electromagnetic wave propagation problem is also considered using two different approaches, namely, a direct time-domain approximation method and a modal analysis technique. For both approaches, a FEM model of the wave propagation problem is obtained by discretizing the spatial domain using the lowest-order Nédélec basis functions. The time domain approximation of this problem is obtained by employing a finite difference (FD) scheme to approximate the second-order temporal derivative in the obtained FEM model. On the other hand, the frequency domain approximation is acquired by truncating the modal expansion solution. Here, it is set forth that the solution to the electromagnetic wave propagation problem can be represented by an expansion of the approximate eigenmodes that are obtained from the associated Maxwell EVP. As a consequence, it is shown by exploiting the numerical test case of the inhomogeneous wave propagation problem that both methodologies lead to accurate approximations which agree well with each other.
Maxwell’s equations, Maxwell source problem, Maxwell eigenvalue problem, Electromagnetic wave propagation problem, Finite elements, Edge elements, Modal analysis
https://hdl.handle.net/11511/102105
Graduate School of Applied Mathematics, Thesis

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Citation Formats
T. Dağlı, “FEM APPROXIMATION OF MAXWELL EQUATIONS: THE SOURCE PROBLEM, EIGENPROBLEM, AND ELECTROMAGNETIC WAVES,” M.S. - Master of Science, Middle East Technical University, 2023.
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