A unified variational formulation for the parabolic-elliptic eddy current equations

Lilian Arnold, Bastian Harrach

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell's equations. The eddy current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). For fixed conductivity, the equations are well studied. However, little rigorous mathematical results are known for the solution's dependence on the conductivity, in particular for the solution's sensitivity with respect to the equation changing from elliptic to parabolic type. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to the conductivity. We then apply our new unified formulation to study the case when the conductivity approaches zero and rigorously linearize the eddy current equations around a nonconducting domain with respect to the introduction of a conducting object.

Original languageEnglish
Pages (from-to)558-576
Number of pages19
JournalSIAM Journal on Applied Mathematics
Issue number2
StatePublished - 2012
Externally publishedYes


  • Eddy current problem
  • Parabolic-elliptic equation
  • Unified variational formulation


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