Convergence Analysis of a Multigrid Method for a Nonlocal Model

Recently, nonlocal models attract the wide interest of scientists. They mainly come from two applied scientific fields: peridynamics and anomalous diffusion. Even though the matrices of the algebraic equation corresponding to the nonlocal models are usually Toeplitz (denote $a_0$ as the principal diagonal element, $a_1$ as the trailing diagonal element, etc). There are still some differences for the models in these two fields. For the model of anomalous diffusion, $a_0/a_1$ is uniformly bounded; most of the time, $a_0/a_1$ of the model for peridynamics is unbounded as the step size $h$ tends to zero. Based on the uniform boundedness of $a_0/a_1$, the convergence of the two-grid method is well established [R. H. Chan, Q.-S. Chang, and H.-W. Sun, SIAM J. Sci. Comput., 19 (1998), pp. 516--529; H. Pang and H. Sun, J. Comput. Phys., 231 (2012), pp. 693--703; M. H. Chen, Y. T. Wang, X. Cheng, and W. H. Deng, BIT, 54 (2014), pp. 623--647]. This paper provides the detailed proof of the convergence of the two-grid method for the nonlocal model of peridynamics. Some special cases of the full multigrid and the V-cycle multigrid methods are also discussed. The numerical experiments are performed to verify the convergence.