Grid and metric optimization in finite difference and finite element methods in magnetic problems

In this paper, the optimization procedures developed recently by the author and coworkers for finite difference and finite element methods are applied to magnetic problems. The grid optimization procedure is used to improve the precision of the results by iteratively improving the grid nodes distribution on the basis of previous solutions of the problem, whereas the metric optimization procedure aims to solve the same problem by using previous solutions in order to iteratively improve the coordinate system in which discretization is to be performed. The improvement in precision obtained for a given number of nodes should allow the same precision to be obtained by means of fewer nodes, thus reducing computer costs, time and memory requirements. In this sense, the author´s experience with the two optimization methods has given satisfactory results with both FDM and FEM, often allowing a strong reduction in costs and computer resources needed to obtain the same precision of results.