Hodge decomposition to solve singular static Maxwell's equations in a non-convex polygon

We are concerned with the singular solution of the static Maxwell equation in a non-convex polygon. Thanks to a Hodge decomposition of the solution on a solenoidal and irrotational parts, one obtains an equivalent formulation to the static problem by solving two Laplace equations. Then a finite element formulation is derived, based on a Nitsche type method. This allows us to solve numerically the static Maxwell equation in domains with reentrant corners, where the solution can be singular. We formulate the method and report some numerical experiments. As a by product, this approach proves its ability to compute the dual singular functions of the Laplacian (see definition below).