A Fully-Discrete Finite Element Approximation for the Eddy Currents Problem

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.