  The Douglas--Rachford (DR) and alternating direction method of multipliers
(ADMM) are two proximal splitting algorithms designed to minimize the sum of
two proper lower semi-continuous convex functions whose proximity operators are
easy to compute. The goal of this work is to understand the local linear
convergence behaviour of DR/ADMM when the involved functions are moreover
partly smooth. More precisely, when the two functions are partly smooth
relative to their respective smooth submanifolds, we show that DR/ADMM (i)
identifies these manifolds in finite time; (ii) enters a local linear
convergence regime. When both functions are locally polyhedral, we show that
the optimal convergence radius is given in terms of the cosine of the
Friedrichs angle between the tangent spaces of the identified submanifolds.
Under polyhedrality of both functions, we also provide condition sufficient for
finite convergence of DR. The obtained results are illustrated by several
concrete examples and supported by numerical experiments.
