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A stabilized [formula omitted] domain decomposition finite element method for time harmonic Maxwell's equations.
- Source :
-
Mathematics & Computers in Simulation . Feb2023, Vol. 204, p556-574. 19p. - Publication Year :
- 2023
-
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 L 2 -based, gradient dependent, triple norm. The order of convergence is O (h) in weighted Sobolev space H w 2 (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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 03784754
- Volume :
- 204
- Database :
- Academic Search Index
- Journal :
- Mathematics & Computers in Simulation
- Publication Type :
- Periodical
- Accession number :
- 159743252
- Full Text :
- https://doi.org/10.1016/j.matcom.2022.08.013