  We propose a domain decomposition method for the efficient simulation of
nonlocal problems. Our approach is based on a multi-domain formulation of a
nonlocal diffusion problem where the subdomains share "nonlocal" interfaces of
the size of the nonlocal horizon. This system of nonlocal equations is first
rewritten in terms of minimization of a nonlocal energy, then discretized with
a meshfree approximation and finally solved via a Lagrange multiplier approach
in a way that resembles the finite element tearing and interconnect method.
Specifically, we propose a distributed projected gradient algorithm for the
solution of the Lagrange multiplier system, whose unknowns determine the
nonlocal interface conditions between subdomains. Several two-dimensional
numerical tests illustrate the strong and weak scalability of our algorithm,
which outperforms the standard approach to the distributed numerical solution
of the problem. This work is the first rigorous numerical study in a
two-dimensional multi-domain setting for nonlocal operators with finite horizon
and, as such, it is a fundamental step towards increasing the use of nonlocal
models in large scale simulations.
