Weak Galerkin finite element methods for H(curl; O) and H(curl, div; O)-elliptic problems

Weak Galerkin finite element methods (WG-FEMs) for H(curl; & omega;) and H(curl, div; & omega;)-elliptic problems are investigated in this paper. The WG method as applied to curl-curl and grad-div problems uses two operators: discrete weak curl and discrete weak divergence, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on the arbitrary shape of polyhedra and, at the same time, is parameter-free. The optimal order of convergence is established for the WG approximations in discrete ������1 norm and ������2 norm. In fact, theoretical convergence analysis holds under low regularity requirements of the analytical solution. Results of numerical experiments that corroborate the theoretical results are also presented.