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Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. (English) Zbl 1094.78008

The paper presents a discontinuous Galerkin method for Maxwell’s equations on unstructured meshes. The computational domain is assumed to be a union of a finite number of polyhedral elements and the electric permittivity and magnetic permeability are assumed to be symmetric and positive definite (in a uniform sense) tensors. Within each element the fields are expanded in a set of vector basis functions. In the general case there is no restriction on these basis functions but for the convergence proofs the authors use piecewise polynomials as basis functions. The method is developed to handle both metallic and absorbing boundary conditions. For metallic cavities it is proved that the scheme preserves a discrete analog of the electromagnetic energy and for cases with absorbing boundaries, the method is \(L^2\)-stable in this energy norm. To achieve this properties the authors use central fluxes and a second order leap-frog scheme for the time integration. It is also shown that a discrete divergence is preserved in a weak sense. For the numerical tests, first and zeroth order elements are used on tetrahedral meshes. The tests are resonance in metallic cavity, scattering by a dielectric sphere and scattering by an aircraft.

MSC:

78M30 Variational methods applied to problems in optics and electromagnetic theory
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
78A40 Waves and radiation in optics and electromagnetic theory
78M25 Numerical methods in optics (MSC2010)
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References:

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