Conditioning and error analysis of nonlocal operators with local boundary conditions

We study the conditioning and error analysis of novel nonlocal operators in ID with local boundary conditions. These operators are used, for instance, in peridynamics (PD) and nonlocal diffusion. The original PD operator uses nonlocal boundary conditions (BC). The novel operators agree with the original PD operator in the bulk of the domain and simultaneously enforce local periodic, antiperiodic, Neumann, or Dirichlet BC. We prove sharp bounds for their condition numbers in the parameter delta only, the size of nonlocality. We accomplish sharpness both rigorously and numerically. We also present an error analysis in which we use the Nystrom method with the trapezoidal rule for discretization. Using the sharp bounds, we prove that the error bound scales like O(h(2)delta(-2)) and verify the bound numerically.