\section{Introduction}
\IEEEPARstart{T}{he} simulation of eddy currents in electrical machines consisting of many steel sheets with the finite element method quickly leads to infeasibly large equation systems. In many machines each sheet in the active zone is exposed to the same field, which allows for a great reduction in computational effort by simulating only a single sheet. However, this reduced problem is still far from trivial. One method to further simplify the problem while maintaining a good approximation of the solution is by spacial decomposition.

The thickness of one sheet is less than a millimeter while the length and width are in the range of meters. A method to treat the two dimensional (2D) cross section and the one dimensional (1D) thickness of the sheet as two coupled problems has been presented in \cite{BottChiamp:02}. It solves the two problems iteratively until convergence is reached. The nature of this coupling has been analyzed in more detail in \cite{PippBelaDlaArkk:10}. 

In \cite{GysSabDul:06} and \cite{HenSteHamGeu:15} different approaches have been presented which isolate the one dimensional problem as a pre-processing step in order to obtain parameters for the two dimensional one.

The 2D/1D multiscale finite element method (MSFEM) presented in \cite{Schoeb:19} uses ideas from the multiscale finite element method to use classic finite element functions for the two dimensional problem while approximating the dependence on the third axis with pre-defined polynomial shape functions, similar to the method presented in \cite{RasDlaFonPippBelaArkkio:11} which is based on trigonometrical shape functions. This enables the solution of the problem within a single iteration while requiring only a mesh for the two dimensional cross section of the sheet. It is also able to include the insulation layers between sheets and correctly treat the edge effect \cite{HollEE:20}.

This paper presents an error estimator for the $\vecT$-formulation of the 2D/1D MSFEM. It is based on flux equilibration and based on the same theory as the error estimator for the $\vecT$-formulation for the MSFEM presented in \cite{Schoeb:21}. In order to fit within the 2D/1D MSFEM framework it has been restructured so both the construction and the evaluation of the estimator require only the two dimensional mesh while being valid in the complete three dimensional domain.

A numerical example shows that the estimator gives a good approximation of the error in both a global and a local sense. This latter property is used to implement adaptive mesh refinement which allows for a high accuracy of the 2D/1D MSFEM solution while requiring significantly less degrees of freedom than uniform mesh refinement.





\section{The $\vecT -\Phi$-Formulation}

We use the $\vecT -\Phi$ formulation for the reference solution of the eddy current problem as described in \cite{Biro:99}. The problem domain $\Omega$ is split into the conducting domain $\Omega_c$, consisting of the steel sheet, and the non-conducting domain $\Omega_0$, consisting of the air regions and the insulation layers. The sheet is assumed to be axis-aligned with the cross-section in the $x-y$ plane and the thickness aligned with the $z$ axis. The total thickness of $\Omega$ is $d=d_{Fe}+d_0$ with the thickness of the sheet $d_{Fe}$ and the thickness of the insulation layer $d_0$.

The magnetic field strength $\vecH$ is written as
\begin{align}
\vecH = \vecT - \nabla\Phi + \vecH_{BS}
\end{align}
with the current vector potential $\vecT\in H(\opcurl ,\Omega_c)$ fulfilling $\opcurl\vecT=\vecJ$, the magnetic scalar potential $\Phi\in H^1(\Omega)$ and a prescribed Biot-Savart field $\vecH_{BS}$. The strong formulation of the eddy current problem in the frequency domain is given as
\begin{align}\label{eq:strong}
\opcurl\rho\opcurl\vecT + i\omega\mu(\vecT - \nabla\Phi + \vecH_{BS}) = \veczero ,
\end{align}
where $\rho=\sigma^{-1}$ is the electric resistivity with the electric conductivity $\sigma$, $\mu$ is the magnetic permeability, $\omega = 2\pi f$ with the frequency $f$ and $i$ is the imaginary unit.

Multiplication with a test function and integration by parts, together with the auxiliary condition $\opdiv\vecB =0$, lead to the weak formulation: Find $\vecT\in H(\opcurl ,\Omega_c)$ and $\Phi\in H^1(\Omega)$ so that
\begin{align}\label{eq:weak3D}
\begin{split}
\int_\Omega \rho\opcurl\vecT\cdot\opcurl\vecv + i\omega\mu(\vecT - \nabla\Phi)\cdot(\vecv - \nabla q) \\
=-\int_\Omega i\omega\mu\vecH_{BS}\cdot(\vecv - \nabla q)
\end{split}
\end{align}
for all $\vecv\in H(\opcurl ,\Omega_c)$ and all $q\in H^1(\Omega)$.






\section{The 2D/1D MSFEM $\vecT$-Formulation}

This paper uses the 2D/1D MSFEM approach for the $\vecT$-formulation which has been described in detail in \cite{Schoeb:19} and \cite{Holl:20}. The three dimensional unknown components $\vecT - \nabla\Phi$ are approximated by
\begin{align}\label{eq:T2D1D}
\vecT_{\text{2D/1D}}= \left( \begin{matrix} 
\phi_0(z)\vecT_{0,x}(x,y) + \phi_2(z)\vecT_{2,x}(x,y) \\ 
\phi_0(z)\vecT_{0,x}(x,y) + \phi_2(z)\vecT_{2,y}(x,y) 
\\ 0
 \end{matrix}\right),
\end{align}
where $\vecT_2\in H(\opcurl_{2D},\Omega_{2D,c})$ and $\vecT_0 = \nabla\Phi_0$ with $\Phi_0\in H^1(\Omega_{2D})$ are defined on the two dimensional projection $\Omega_{2D}$ of $\Omega$. Here and in the following, coordinates $x$, $y$ or $z$ in the index denote the individual components of a vector-valued function. The shape functions $\phi_0$ and $\phi_2$ are predefined piecewise polynomial of order $0$ and $2$, respectively, see also 