An upscaled DGTD method for time-domain electromagnetics

Abstract : In this work, we address time-dependent electromagnetic wave propagation problems with strong multiscale features with application to nanophotonics, where problems usually involve complex mul-tiscale geometries, heterogeneous materials, and intense, localized electromagnetic fields. Nanopho-tonics simulations require very fine meshes to incorporate the influence of geometries as well as high order polynomial interpolations to minimize dispersion. Our goal is to design a family of innovative high performance numerical methods perfectly adapted for the simulation of such multiscale problems. For that purpose we extend the Multiscale Hybrid-Mixed (MHM) finite element method, originally proposed for the Laplace problem in [1], to the solution of 2d and 3d transient Maxwell equations with heterogeneous media. The MHM method arises from the decomposition of the exact electromagnetic fields in terms of the solutions of locally independent Maxwell problems. Those problems are tied together with an one field formulation on top of a coarse mesh skeleton. The multiscale basis functions, which are responsible for upscaling, are also driven by local Maxwell problems. A high order Discontinuous Galerkin method (see [2]) in space combined with a second-order explicit leapfrog scheme in time discretizes the local problems. This makes the MHM method effective and parallelizable, and yields a staggered algorithm within a divide-and-conquer framework. In this talk we will present the general formulation of this MHM-DGTD method and presen some preliminary numerical results in 2d. REFERENCES 1. C. Harder, D. Paredes, and F. Valentin. A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients. J. Comput. Phys., 245:107-130, 2013. 2. S. Descombes, C. Durochat, S. Lanteri, L. Moya, C. Scheid, and J. Viquerat. Recent advances on a DGTD method for time-domain electromagnetics. Photonics and Nanostructures-Fundamentals and Applications, 11(4):291-302, 2013.
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A Gobé, S. Lanteri, Raphaël Léger, C Scheid, F. Valentin. An upscaled DGTD method for time-domain electromagnetics. Progress In Electromagnetics Research Symposium - PIERS 2018, Aug 2018, Toyama, Japan. ⟨hal-01974085⟩

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