# A discontinuous Galerkin method for the time harmonic eddy current problem

## TL;DR

This paper presents a novel discontinuous Galerkin method for solving the time-harmonic eddy current problem, combining vector and scalar magnetic field approximations with weak transmission conditions, ensuring stability and accuracy.

## Contribution

The paper introduces a new DG scheme for magnetic field problems that integrates vector and scalar field approximations with weakly enforced transmission conditions, providing stability and error estimates.

## Key findings

- The method is proven to be uniformly stable.
- Quasi-optimal error estimates are established.
- The scheme effectively couples vector and scalar magnetic field problems.

## Abstract

We introduce and analyze a discontinuous Galerkin method for a time-harmonic eddy current problem formulated in terms of the magnetic field. The scheme is obtained by putting together a DG method for the approximation of the vector field variable representing the magnetic field in the conductor and a DG method for the Laplace equation whose solution is a scalar magnetic potential in the insulator. The transmission conditions linking the two problems are taken into account weakly in the global discontinuous Galerkin scheme. We prove that the numerical method is uniformly stable and obtain quasi-optimal error estimates in the DG-energy norm.