RESIDUAL-BASED A POSTERIORI ESTIMATORS FOR THE A - φ MAGNETODYNAMIC HARMONIC FORMULATION OF THE MAXWELL SYSTEM

Abstract

This paper is devoted to the derivation of an a posteriori residual-based error estimator for the A-φ magnetodynamic harmonic formulation of the Maxwell system. The weak continuous and discrete formulations are established, and the well-posedness of both of them is addressed. Some useful analytical tools are derived. Among them, an ad hoc Helmholtz decomposition is proven, which allows to pertinently split the error. Consequently, an a posteriori error estimator is obtained, which is proven to be reliable and locally efficient. Finally, numerical tests confirm the theoretical results.

A posteriori error estimator for harmonic A‐φ formulation

“…where A belongs to a given space denoted X 0 (D) so that the gauge condition is ensured, and where w belongs toH 1 (D c ) (Creusé et al, 2012). Moreover, A 0 and w 0 are the test functions belonging to the same spaces as A and w, respectively.…”

“…From the residual term it is possible to develop the residual based error estimator (Creusé et al, 2012).…”

“…For the harmonic A-w formulation from the definitions of the numerical errors (equations (9)-(15)) the local error estimator on a tetrahedron T can be developed under the form (Creusé et al, 2012):…”

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

“…In this work, we are particularly interested in explicit residual-based error estimators devoted to the approximation of Maxwell's equations in the low frequency regime by harmonic formulations, where the quasi-static approximation occurs [3][4][5][6].…”

Existence results for the A & φ magnetodynamic formulation of the Maxwell system

“…[4] Concerning the harmonic formulation of some Maxwell problems, several contributions have been proposed in the last decade. In that case, since no time derivatives are involved, we have only to deal with spatial problems for which existence results are easier to obtain (see for instance Theorem 2.1 in [5]). …”