Nonlocal continuum models are used in many scientific and engineering applications where the material response and dynamics depend on the micro-structure. Such models differ from the classical, local, models in the fact that interactions can occur at distance, without contact; for this reason they are used for accurately resolve small scale features such as crack tips or dislocations that can affect the global material behavior. However, nonlocal models are often computationally too expensive, sometimes even intractable. Therefore, methods for the coupling of nonlocal and local models have been proposed for efficient and accurate solutions; these methods employ nonlocal models in small parts of the domain and use local, macroscopic, models elsewhere.
We propose an optimization-based coupling method for nonlocal diffusion problems; we split the domain in a nonlocal and local domain such that they feature a non-zero intersection and we minimize the difference between the nonlocal and local solutions in the overlapping regions tuning their values on the common boundaries and volumes. We formulate the problem as a control problem where the states are the solutions of the nonlocal and local equations, the controls are the nonlocal volume constraint and the local boundary condition, and the objective of the optimization is a matching functional for the state variables in the intersection of the domains. The problem is treated in a variational sense and its analysis is conducted using the nonlocal vector calculus, a recently developed tool that allows one to solve a nonlocal problem similarly as the local counterpart. Specifically, we show that the coupling problem is well-posed, we study the modeling error and, for finite element discretizations, we analyze the approximation error. Furthermore, we present numerical results in a one-dimensional setting; though preliminary, our tests show the consistency of the method, illustrate the theoretical results and provide the basis for realistic simulations.