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

Abstract

The multiscale finite element method (MSFEM) 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 using a coarse mesh that does not resolve each sheet and constructs the local fields using predefined micro-shape functions. This article presents for the first time an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> error estimator for the MSFEM, 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 linear and nonlinear cases.