非局部问题的多重网格方法：非对角占优的Toeplitz-plus-tridiagonal系统
A Multigrid method for nonlocal problems: non-diagonally dominant Toeplitz-plus-tridiagonal systems
AI总结：
本文研究了非局部问题的多重网格方法，针对非对角占优的Toeplitz-plus-tridiagonal系统，提出了基于谱分析的简单限制和细化算子，以处理非Toeplitz且非弱对角占优的代数系统，并验证了多重网格方法的收敛性。
摘要：
非局部问题已被用于建模非常不同的应用科学现象，当考虑莱维过程和随机界面时，会涉及分数拉普拉斯算子。本文研究了在有界域上的非局部问题，其中得到的系统刚度矩阵是Toeplitz-plus-tridiagonal且远离对角占优，这发生在处理线性有限元近似时。通过利用刚度矩阵的弱对角占优Toeplitz性质，两种网格方法的最优收敛性得以确立[Fiorentino和Serra-Capizzano, SIAM J. Sci. Comput., 17(1996), pp. 1068--1081; Chen和Deng, SIAM J. Matrix Anal. Appl., 38(2017), pp. 869--890]；当刚度矩阵远离弱对角占优时，关于如何定义粗化和插值算子的最佳方法仍存在问题[Stüben, J. Comput. Appl. Math., 128(2001), pp. 281--309]。在本文中，利用对涉及矩阵的谱分析得到的谱指示，采用简单的（传统）限制算子和细化算子来处理一般的代数系统，这些系统既不是Toeplitz也不是弱对角占优，分别对应于分数拉普拉斯核和常数核。本文致力于为这种情况下多重网格方法的收敛性提供详细的证明。此外，还讨论了常数核情况下的全多重网格方法的收敛性。数值实验用于验证仅通过快速傅里叶变换以O(N log N)复杂度进行收敛性验证，其中N是网格点的数量。
英文摘要：
The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the Lévy 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 [Stüben, {\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.
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