The Discontinuous Galerkin Method Applied to Problems in Electromagnetism
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The discontinuous Galerkin method (DGM) is applied to a number of problems in computational electromagnetics. This is achieved by obtaining numerical solutions to Maxwell's equations using the DGM. The aim of these simulations is to highlight the strengths of the method while showing its resilience in handling problems other schemes may not be able to accurately model. Although no method will ever be the best choice for every problem in electromagnetics, the discontinuous Galerkin method is able to accurately approximate any problem, although the computational costs can make the scheme impractical for some. Like other time domain schemes, the DGM becomes inefficient on large domains where the solution contains small wavelengths. We demonstrate that all of the different types of boundary conditions in electromagnetic wave propagation can be implemented into the DGM. Reflection and transmission boundaries fit easily into the framework, whereas perfect absorption requires a more advanced technique known as the perfectly matched layer. We begin by simulating mirrors with several different geometries, and analyze how the DGM method performs, and how it offers a more complete evaluation of the behavior in this problem than some other methods. Since Maxwell's equations describe the macroscopic features of electromagnetics, our simulations are able to capture the wave features of electromagnetics, such as interference and diffraction. We demonstrate this by accurately modelling Young's double slit experiment, a classic experiment which features well understood interference and diffraction phenomena. We also extend the basic electromagnetic wave propagation simulations to include situations where the waves travel into new media. The formulation of the DGM for Maxwell's equations allows the numerical solutions to accurately resolve the features at the interface of two media as predicted by the Fresnel coefficients. This allows the DGM to model lenses and other sources of refraction. We predict that the DGM will become an increasingly valuable method for computational electromagnetics because of its wide range of applicability as well as the lack of undesirable features in the numerical solutions. Furthermore, the only limiting factor for applying DGM, its computational cost, will become less influential as computing power continues to increase, allowing us to apply the DGM to an increasing set of applications.