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

• Published in: Mathematics and computers in simulation Vol. 204; p. 556
• Main Authors: Asadzadeh, M., Beilina, L.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 2023
• Subjects: [formula omitted] finite elements; A posteriori estimate; A priori estimate; Beräkningsmatematik; Computational Mathematics; Computer Science; Convergence; discontinuous galerkin methods; Mathematics; P1 finite elements; priori estimate; Stability; Time harmonic Maxwell?s equations; Time harmonic Maxwell’s equations; A priori estimate; Stability; P1 finite elements; A posteriori estimate; Convergence
• ISSN: 0378-4754; 1872-7166
• DOI: 10.1016/j.matcom.2022.08.013

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), Di Pietro and Ern (2012). 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.