Adaptive finite element analysis of 2-D static and steady-state electromagnetic problems

The adaptive Þnite element methods developed by the authors for a class of 2-D problems in computational electromagnetics are presented. The cases covered are: magnetostatic, electrostatic, DC conduction and linear AC steady-state eddy currents, both plane and axialsymmetric. Theory is developed in the linear case only, but the resulting methods are applied on a heuristic basis also to non-linear cases and prove able to cope with them. These methods make use of an element-by-element error estimation and two di¤erent h-reÞnement techniques to adapt meshes made up of Þrst or second-order triangular elements. Element-byelement error estimation has two main advantages: it is well suited to cope with the intricate geometries of electromagnetic devices and, at the same time, very cheap from a computational viewpoint. The local error is estimated by approximately solving a di¤erential problem in each element. Theory on which this error estimation method is grounded is developed in full details and the underlying assumptions are pointed out. The presence in electromagnetic problems of surface currents and interfaces between di¤erent materials poses new problems in the error estimation with respect to other applications in which similar methods are used. The h-reÞnement technique can be selected between the bisection method and the centroid method followed by a Delaunay step. Accuracy and e¦ciency of the proposed method is discussed on the basis of previous work of the authors and further numerical experiments. Applications to realistic models showing good performances of the proposed methods are reported.