An Equilibrated Error Estimator for the Multiscale Finite Element Method of a 2-D Eddy Current Problem

Abstract

The multiscale finite element method is a valuable tool to solve the eddy current problem in laminated materials consisting of many iron sheets, which would be prohibitively expensive to resolve in a finite element mesh. It allows to use a coarse mesh which does not resolve each sheet and constructs the local fields using predefined micro-shape functions. This paper presents for the first time an a-posteriori error estimator for the multiscale finite element method which considers the error with respect to the exact solution. It is based on flux equilibration and a modification of the theorem of Prager and Synge and provides an upper bound for the error that does not include generic constants. Numerical examples show a good performance in both the linear and the nonlinear case.

An Equilibrated Error Estimator for the 2-D/1-D MSFEM T-Formulation of the Eddy Current Problem

“…A variant of (9) for the two dimensional scalar Tformulation has been proven in [8] and for the vector-valued magnetostatic case in [12]. The proof of ( 9) is analogous.…”

“…This paper presents an error estimator for the T-formulation of the 2D/1D MSFEM. It is based on flux equilibration and similar to the error estimator for the T-formulation for the MSFEM presented in [8]. The theory has been restructured in order to fit within the 2D/1D MSFEM framework.…”

A Hierarchical Error Estimator for the MSFEM for the Eddy Current Problem in 3-D

“…where the asterisk denotes the complex conjugate. A variant of ( 9) for the two dimensional scalar Tformulation has been proven in [8] and for the vector-valued magnetostatic case in [12]. The proof of ( 9) is analogous.…”

“…This paper presents an error estimator for the T-formulation of the 2D/1D MSFEM. It is based on flux equilibration and based on the same theory as the error estimator for the Tformulation for the MSFEM presented in [8]. In order to fit within the 2D/1D MSFEM framework it has been restructured so both the construction and the evaluation of the estimator require only the two dimensional mesh while being valid in the complete three dimensional domain.…”