  We propose a primal-dual parallel proximal splitting method for solving
domain decomposition problems for partial differential equations. The problem
is formulated via minimization of energy functions on the subdomains with
coupling constraints which model various properties of the solution at the
interfaces. The proposed method can handle a wide range of linear and nonlinear
problems, with flexible, possibly nonlinear, transmission conditions across the
interfaces. Strong convergence in the energy spaces is established in this
general setting, and without any additional assumption on the energy functions
or the geometry of the problem. Several examples are presented.
