  The nonlocal problems have been used to model very different applied
scientific phenomena, which involve the fractional Laplacian when one looks at
the L\'{e}vy processes and stochastic interfaces. This paper deals with the
nonlocal problems on a bounded domain, where the stiffness matrices of the
resulting systems are Toeplitz-plus-tridiagonal and far from being diagonally
dominant, as it occurs when dealing with linear finite element approximations.
By exploiting a weakly diagonally dominant Toeplitz property of the stiffness
matrices, the optimal convergence of the two-grid method is well established
[Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. Comput.}, {17} (1996), pp.
1068--1081; Chen and Deng, {\em SIAM J. Matrix Anal. Appl.}, {38} (2017), pp.
869--890]; and there are still questions about best ways to define coarsening
and interpolation operator when the stiffness matrix is far from being weakly
diagonally dominant [St\"{u}ben, {\em J. Comput. Appl. Math.}, {128} (2001),
pp. 281--309]. In this work, using spectral indications from our analysis of
the involved matrices, the simple (traditional) restriction operator and
prolongation operator are employed in order to handle general algebraic systems
which are {\em neither Toeplitz nor weakly diagonally dominant} corresponding
to the fractional Laplacian kernel and the constant kernel, respectively. We
focus our efforts on providing the detailed proof of the convergence of the
two-grid method for such situations. Moreover, the convergence of the full
multigrid is also discussed with the constant kernel. The numerical experiments
are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$
complexity by the fast Fourier transform, where $N$ is the number of the grid
points.
