  This article presents a new primal-dual weak Galerkin finite element method
for the div-curl system with tangential boundary conditions and low-regularity
assumptions on the solution. The numerical scheme is based on a weak
variational form involving no partial derivatives of the exact solution
supplemented by a dual or ajoint problem in the general context of the weak
Galerkin finite element method. Optimal order error estimates in $L^2$ are
established for solution vector fields in $H^\theta(\Omega),\ \theta>\frac12$.
The mathematical theory was derived on connected domains with general
topological properties (namely, arbitrary first and second Betti numbers).
Numerical results are reported to confirm the theoretical convergence.
