Abstract
The aim of this article is to develop an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> hierarchical error estimator for the multiscale finite element method (MSFEM). The MSFEM is used to solve the linear eddy current problem in a stack of iron sheets. It allows for a calculation of the solution without having to resolve each sheet in the finite element mesh. A hierarchical local error estimator for nodal elements and edge elements is adapted to the multiscale setting. The estimator allows for adaptive p-refinement on the coarse multiscale mesh. Numerical examples show an increased order of convergence compared to uniform refinement. The proposed error estimator increases the efficiency of the MSFEM, requiring even fewer degrees of freedom. It is the first error estimator presented for the 3-D MSFEM for the eddy current problem.
Highlights
T HE simulation of eddy currents in a laminated iron core is a challenging task, as resolving each sheet in the finite element mesh leads to an unfeasible number of unknowns
The developed estimator is based on a hierarchical error estimator that has been proposed for the H 1 in [15], for the H in [16] and included in the analysis presented in [17]
That, due to the nature of multiscale finite element method (MSFEM), this leads to only a slight increase in degrees of freedom because many sheets can be covered by a single finite element
Summary
T HE simulation of eddy currents in a laminated iron core is a challenging task, as resolving each sheet in the finite element mesh leads to an unfeasible number of unknowns. This article focuses on the MSFEM, which has been presented in [2] for the eddy current problem in 3-D, using a formulation based on the magnetic vector potential A [5]. Equilibrated estimators are more theoretically involved but allow for estimated errors that do not include generic constants [9] Both types of estimators have been compared for the magnetostatic problem [10] and the eddy current problem [11]. An important application for local error estimators is adaptive mesh refinement [12]. This allows to increase the quality of the solution using significantly fewer unknowns than a uniform refinement [13]. It is demonstrated that the estimator correctly identifies the local behavior of the solution and that it increases the rate of convergence of the numerical error with respect to the used degrees of freedom
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