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

The aim of this article is to develop an a posteriori 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.