A stabilized P1 domain decomposition finite element method for time harmonic Maxwell’s equations

M. Asadzadeh and L. Beilina

Mathematics and Computers in Simulation (MATCOM), 2023, vol. 204, issue C, 556-574

Abstract: One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell’s equations is to render their hyperbolic character to elliptic form. This paper is devoted to the study of a stabilized linear, domain decomposition, finite element method for the time harmonic Maxwell’s equations, in a dual form, obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. The discrete problem is coercive in a symmetrized norm, equivalent to the discrete norm of the model problem. This yields discrete stability, which together with continuity guarantees the well-posedness of the discrete problem, cf Arnold et al. (2002) [3], Di Pietro and Ern (2012) [45]. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L2-based, gradient dependent, triple norm. The order of convergence is O(h) in weighted Sobolev space Hw2(Ω), and hence optimal. Here, the weight w≔w(ɛ,s) where ɛ is the dielectric permittivity function and s is the Laplace transformation variable. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residuals of the computed solution over the domain and at the boundary (involving the relevant jump terns) and hence independent of the unknown exact solution. The a posteriori approach is used, e.g. in constructing adaptive algorithms for the computational purposes, which is the subject of a forthcoming paper. Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme.

Keywords: Time harmonic Maxwell’s equations; P1 finite elements; Stability; A priori estimate; A posteriori estimate; Convergence (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:204:y:2023:i:c:p:556-574

DOI: 10.1016/j.matcom.2022.08.013

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