A singular field method for Maxwell's equations: Numerical aspects for 2D magnetostatics

The present paper deals with the solution of Maxwell-type problems by means of nodal H-1-conforming finite elements. In a nonconvex piecewise regular domain surrounded by a perfect conductor, such a discretization cannot in general approximate the singular behavior of the electromagnetic field near "reentrant" corners or edges. The singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior. The latter are deduced from the singular functions associated with the scalar Laplace operator. The theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting, but the study can be easily extended to numerous Maxwell-type problems. The numerical implementation of both variants is studied for a domain containing a single reentrant corner.