We consider a finite element discretization for the dual Rudin-Osher-Fatemi model using a Raviart-Thomas basis for H-0 (div; Omega). Since the proposed discretization has a splitting property for the energy functional, which is not satisfied for existing finite difference-based discretizations, it is more adequate for designing domain decomposition methods. In this paper, a primal domain decomposition method is proposed which resembles the classical Schur complement method for the second order elliptic problems, and it achieves O(1/n(2)) convergence. A primal-dual domain decomposition method based on the method of Lagrange multipliers on the subdomain interfaces is also considered. Local problems of the proposed primal-dual domain decomposition method can be solved at a linear convergence rate. Numerical results for the proposed methods are provided.