An Error Estimator for Multiscale FEM for the Eddy-Current Problem in Laminated Materials

Abstract

This paper develops an error estimator for a known multiscale method, which is used to solve the eddy-current problem using the time-harmonic single component current vector potential in 2-D. The multiscale method allows for the solution of the problem in a laminated domain without needing to resolve each laminate in the mesh, which would require a prohibitively large number of degrees of freedom in the finite-element system. The error estimator is based on a flux equilibration technique, which has so far been presented for a more restricted class of problems and has the advantage of being efficient as well as reliable with a generic constant equal to 1. It can be shown to be extendable to the equations arising in the used multiscale method. Its local nature allows for the construction of an efficient adaptive mesh refinement. Using these findings, the multiscale method can be improved to obtain a better rate of convergence with respect to the number of degrees of freedom.