Analysis of eddy current formulations in two-dimensional domains with cracks

Lohrengel, Stephanie; Nicaise, Serge

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 1, pp. 141-170.

In this paper, the eddy current problem in a two-dimensional conductor containing a crack is studied. The decomposition of the electric field into a piecewise regular part and a singular part deriving from scalar potentials localized at the crack tip and at the crack mouth is proved. At the crack mouth, the electric field is shown to have standard singularities inside the conductor, but presents a singularity outside the conductor that does not belong to the classical $L^2$-space. Well-posedness of the $𝐄$-based model and the $𝐀-\psi$-formulation of combined potentials are proved and an un-gauged discretization of the latter formulation is discussed.

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