Preprint 2026-36
Ramiro Acevedo, Rommel Bustinza, Christian Gómez:
A hybridizable discontinuous Galerkin method for the time-dependent eddy current problem in an electric field formulation
Abstract:
In this work we propose and analyze a hybridizable discontinuous Galerkin (HDG) method for the time-dependent eddy current problem in an electric field formulation, posed in a domain consisting of a conductor and a dielectric region. The divergence-free condition in the dielectric and the zero-flux condition on the interface between both media are imposed by means of a Lagrange multiplier. The flux condition only adds one global scalar unknown, so that the static condensation remains local. The time discretization is performed with the backward Euler scheme. We prove that the fully discrete problem is well posed and derive error estimates of order $\mathcal O(h^{k+1/2}+\Delta t)$ for polynomial degree $k\ge0$. To the best of our knowledge, these are the first error estimates for an HDG method applied to the transient eddy current problem. Numerical experiments show first order in $\Delta t$ and, for $k=0,1,2$, computed spatial orders are better than the proved ones. Moreover, we illustrate the robustness of the method on a model problem of magnetic shielding by eddy currents.
