Discontinuous Least-Squares finite element method for the Div-Curl problem

In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method.