Coupling of Local and Nonlocal Problems Using Local Boundary Conditions

We present a novel coupling method for local and nonlocal diffusion problems in 1D. Unlike other methods, our coupling method exclusively uses local boundary conditions. This is possible because our nonlocal operators enforce them by construction. Leveraging this advantageous property, we construct a seamless coupling that is remarkably natural. The utilization of local boundary conditions allows for the transfer of well-established numerical methods from local problems to nonlocal ones. Our local-to-nonlocal coupling method is inspired by the domain decomposition method, which we would like to transfer to nonlocal problems. The main result of our study is the construction of a local-to-nonlocal coupling method with a quantifiable $O(h)$ convergence that holds for an arbitrary solution. For discretization of the local and nonlocal problems, the finite element method and the Galerkin projection are employed, respectively. We verify our convergence rate with extensive numerical experiments.