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The curl-div system: theory and finite element approximation. (English) Zbl 1445.78010

Langer, Ulrich (ed.) et al., Maxwell’s equations. Analysis and numerics. Contributions from the workshop on analysis and numerics of acoustic and electromagnetic problems, RICAM, Linz, Austria, October 17–22, 2016. Berlin: De Gruyter. Radon Ser. Comput. Appl. Math. 24, 1-43 (2019).
Summary: We first propose and analyze two variational formulations of the curl-div system that rewrite it as a saddle-point problem. Existence and uniqueness results are then an easy consequence of this approach. Second, introducing suitable constrained Hilbert spaces, we devise other variational formulations that turn out to be useful for numerical approximation. Curl-free and divergence-free finite elements are employed for discretizing the problem, and the corresponding finite element solutions are shown to converge to the exact solution. Several numerical tests are also included, illustrating the performance of the proposed approximation methods.
For the entire collection see [Zbl 1420.78002].

MSC:

78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78M30 Variational methods applied to problems in optics and electromagnetic theory
35A15 Variational methods applied to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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