  We introduce a mathematically rigorous formulation for a nonlocal interface
problem with jumps and propose an asymptotically compatible finite element
discretization for the weak form of the interface problem. After proving the
well-posedness of the weak form, we demonstrate that solutions to the nonlocal
interface problem converge to the corresponding local counterpart when the
nonlocal data are appropriately prescribed. Several numerical tests in one and
two dimensions show the applicability of our technique, its numerical
convergence to exact nonlocal solutions, its convergence to the local limit
when the horizons vanish, and its robustness with respect to the patch test.
