Abstract
The eddy current model is obtained from Maxwell’s equations by neglecting the displacement currents in the Amp`ere-Maxwell’s law and it is commonly used in many problems in sciences, engineering and industry (e.g, in induction heating, electromagnetic braking, and power transformers). The so-called “A, V −A potential formulation” (B´ır´o & Preis [1]) is nowadays one of the most accepted formulations to solve the eddy current equations numerically, and B´ır´o & Valli [2] have recently provided its well-posedness and convergence analysis for the time-harmonic eddy current problem. The aim of this paper is to extend the analysis performed by B´ır´o & Valli to the general transient eddy current model. We provide a backward-Euler fully-discrete approximation based on nodal finite elements and we show that the resulting discrete variational problem is well posed. Furthermore, error estimates that prove optimal convergence are settled.
Highlights
In applications related to electrical power engineering the displacement currents existing in a metallic conductor are negligible compared to the conduction current
Among the numerical methods used to approximate eddy current equations, the finite element method (FEM) and methods combining FEM and boundary element method (FEM-BEM) are the most extended, see, for instance, the recent book by Alonso & Valli [4] for a survey on this subject including a large list of references
We provide a backward-Euler fully-discrete approximation based on nodal finite elements to approximate the solution of the resultant model
Summary
In applications related to electrical power engineering (see for instance [1]) the displacement currents existing in a metallic conductor are negligible compared to the conduction current. The aim of this paper is to analyze a finite element fully-discrete approximation for the time-dependent eddy current problem in a bounded domain, based in a formulation in terms of a vector magnetic potential and an electric scalar potential. This fully-discrete scheme solves in each time an elliptic problem and we use the techniques used in [14] and [17] to prove the ellipticity of its bilinear form By using this ellipticity we define projection operators to the discrete finite element subspaces and obtain quasi-optimal error estimates, which allows us to approximate the typical physical variables of interest of the eddy current problem: the eddy currents in the conductor and the magnetic induction in the computational domain.
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