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.

中文翻译：

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非局部问题的多重网格方法：非对角占优或 Toeplitz-Plus-Tridiagonal 系统

非局部问题已被用于模拟非常不同的应用科学现象，当人们查看 L\'{e}vy 过程和随机接口时，这些现象涉及分数拉普拉斯算子。本文处理有界域上的非局部问题，其中所得系统的刚度矩阵是 Toeplitz-plus-tridiagonal 并且远非对角主导，因为它在处理线性有限元近似时发生。通过利用刚度矩阵的弱对角优势 Toeplitz 特性，很好地建立了双网格方法的最佳收敛 [Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. 计算。}，{17}（1996 年），第 1068--1081 页；Chen 和 Deng，{\em SIAM J. Matrix Anal。应用程序}，{38}（2017 年），第 869--890 页]；