  We consider a finite element discretization for the dual Rudin--Osher--Fatemi
model using a Raviart--Thomas basis for $H_0 (\mathrm{div};\Omega)$. Since the
proposed discretization has 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 in linear convergence rate. Numerical results for the proposed methods
are provided.
