Residual‐based a posteriori estimators for the potential formulations of electrostatic and time‐harmonic eddy current problems with voltage or current excitation

Abstract

SUMMARYIn this paper, we consider some potential formulations of electrostatic as well as time-harmonic eddy current problems with voltage or current excitation sources. The well-posedness of each formulation is first established. Then, the reliability of the corresponding residual-based a posteriori estimators is derived in the context of the finite element method approximation. Finally, the implementation in an industrial code is performed, and the obtained theoretical results are illustrated on an academic and on an industrial benchmark.

Adaptive Finite Element Method for the Maxwell Eigenvalue Problem

“…We now estimate the second term in (18). Recalling the mixed problem (16) defining ζ h and r h , we denote by ζ H ∈ Σ H and r H ∈ Q H the corresponding solution on the mesh T H with g = −(curl(σ h − σ H ).…”

“…Putting together the estimates of the three terms in (18), and using the a priori error estimate for the eigenvalue problem (see, for instance [10] or [7]), we obtain by a suitable definition of ρ 1 (H) (22) ζ h…”

“…A posteriori error estimates for Maxwell's equations have been studied by several authors for the source problem (see, in particular [32,3,37,19,36,20,38,22,14,23,42,15,21,18] and the references therein). The eigenvalue problem has been studied only recently in [13,12] where residual type error indicators are considered and proved to be equivalent to the actual error in the standard framework of efficiency and reliability.…”

Goal-oriented error estimation based on equilibrated flux reconstruction for the approximation of the harmonic formulations in eddy current problems

“…Consequently, some a posteriori error estimators have been developed in order to provide a global upper bound of the numerical error, as well as some local lower bounds, very useful to drive a mesh-refinement strategy. We refer to [11,26,5] for residual estimators and to [12,9] for equilibrated ones, allowing in this second case to obtain a sharp upper bound of the error without any unknown multiplicative constant. All these estimators have been tested in several configurations, from academic to more industrial ones (see e.g.…”