# A Multigrid Method for Nonlocal Problems: Non--Diagonally Dominant or Toeplitz-Plus-Tridiagonal Systems

## Abstract

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 or far from being diagonally dominant as 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 in [Fiorentino and Serra-Capizzano, SIAM J. Sci. Comput., 17 (1996), pp. 1068--1081; Chen and Deng, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 869--890], and there are still questions about the best ways to define the coarsening and interpolation operators when the stiffness matrix is far from being weakly diagonally dominant [Stüben, 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 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 $\mathcal{O}(N {log} N)$ complexity by the fast Fourier transform, where $N$ is the number of the grid points.

## References

1.
G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), pp. 472--495.
2.
B. Aksoylu and Z. Unlu, Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces, SIAM J. Numer. Anal., 52 (2014), pp. 653--677.
3.
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi, and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Math. Surveys Monogr. 165, AMS, Providence, RI, 2010.
4.
A. Aricò and M. Donatelli, A V-cycle multigrid for multilevel matrix algebras: Proof of optimality, Numer. Math., 105 (2007), pp. 511--547.
5.
A. Aricò, M. Donatelli, and S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 186--214.
6.
R. E. Bank and C. C. Douglas, Sharp estimates for multigrid rates of convergence with general smoothing and acceleration, SIAM J. Numer. Anal., 22 (1985), pp. 617--633.
7.
P. Bates, On Some Nonlocal Evolution Equations Arising in Materials Science, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., H. Brunner, X. Zhao, and X. Zou, eds., AMS, Providence, RI, 2006, pp. 13--52.
8.
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996.
9.
M. Bolten, M. Donatelli, T. Huckle, and C. Kravvaritis, Generalized grid transfer operators for multigrid methods applied on Toeplitz matrices, BIT, 55 (2015), pp. 341--366.
10.
J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp., 49 (1987), pp. 311--329.
11.
J. H. Bramble, J. E. Pasciak, J. P. Wang, and J. C. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comp., 57 (1991), pp. 23--45.
12.
W. L. Briggs, V. E. Henson, and S. F. Mccormick, A Multigrid Tutorial, SIAM, Philadelphia, 2000.
13.
R. H. Chan, Q. S. Chang, and H. W. Sun, Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput., 19 (1998), pp. 516--529.
14.
R. H. Chan and X. Q. Jin, An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.
15.
L. Chen, R. H. Nochetto, E. Otárola, and A. J. Salgado, Multilevel methods for nonuniformly elliptic operators and fractional diffusion, Math. Comp., 85 (2016), pp. 2583--2607.
16.
M. H. Chen, W. Y. Qi, and Y. T. Wang, Uniform convergence of V-cycle multigrid finite element method for one-dimensional time-dependent fractional problem, Appl. Math. Lett., 98 (2019), pp. 49--56.
17.
M. H. Chen and W. H. Deng, High order algorithms for the fractional substantial diffusion equation with truncated Levy flights, SIAM J. Sci. Comput., 37 (2015), pp. A890--A917.
18.
M. H. Chen and W. H. Deng, Convergence analysis of a multigrid method for a nonlocal model, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 869--890.
19.
M. H. Chen, W. H. Deng, and S. Serra-Capizzano, Uniform convergence of V-cycle multigrid algorithms for two-dimensional fractional Feynman-Kac equation, J. Sci. Comput., 74 (2018), pp. 1034--1059.
20.
M. H. Chen, Y. T. Wang, X. Cheng, and W. H. Deng, Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation, BIT, 54 (2014), pp. 623--647.
21.
M. D'Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl., 66 (2013), pp. 1254--1260.
22.
M. Donatelli, M. Mazza, and S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys., 307 (2016), pp. 262--279.
23.
Q. Du, M. Gunzburger, R. Lehoucq, and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 56 (2012), pp. 676--696.
24.
G. Fiorentino and S. Serra-Capizzano, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions, SIAM J. Sci. Comput., 17 (1996), pp. 1068--1081.
25.
W. Hackbusch, Multigrid Methods and Applications, Springer-Verlag, Berlin, 1985.
26.
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 2013.
27.
Y. H. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference-quadrature approach, SIAM J. Numer. Anal., 52 (2014), pp. 3056--3084.
28.
Y. J. Jiang and X. J. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), pp. 374--392.
29.
M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), pp. 7--51.
30.
T. Mengesha and Q. Du, Analysis of a scalar nonlocal peridynamic model with a sign changing kernel, Discrete Contin. Dyn. Syst. B, 18 (2013), pp. 1415--1437.
31.
H. Moghaderi, M. Dehghan, M. Donatelli, and M. Mazza, Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations, J. Comput. Phys., 350 (2017), pp. 992--1011.
32.
M. Ng, S. Serra-Capizzano, and C. Tablino-Possio, Multigrid method for symmetric sinc-Galerkin systems, Numer. Linear Algebra Appl., 12 (2005), pp. 261--269.
33.
H. Pang and H. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), pp. 693--703.
34.
J. Pang, R. Ke, M. Ng, and H. Sun, Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Sci. Comput., 36 (2014), pp. A2698--A2719.
35.
J. Ruge and K. Stüben, Algebraic Multigrid, in Multigrid Methods, S. McCormick, ed., SIAM, Philadelphia, 1987, pp. 73--130.
36.
37.
S. Serra-Capizzano, On the extreme spectral properties of Toeplitz matrices generated by $L1$ functions with several minima/maxima, BIT, 36 (1996), pp. 135--142.
38.
S. Serra-Capizzano, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270 (1998), pp. 109--129.
39.
S. Serra-Capizzano, Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences, Numer. Math., 92 (2002), pp. 433--465.
40.
S. Serra-Capizzano and C. Tablino-Possio, Two-grid methods for Hermitian positive definite linear systems connected with an order relation, Calcolo, 51 (2014), pp. 261--285.
41.
S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids, 48 (2000), pp. 175--209.
42.
K. Stüben, A review of algebraic multigrid, J. Comput. Appl. Math., 128 (2001), pp. 281--309.
43.
X. C. Tian and Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal., 51 (2013), pp. 3458--3482.
44.
W. Wang and H. Tian, A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model, J. Comput. Phys., 231 (2012), pp. 7730--7738.
45.
J. Xu, An introduction to multilevel methods, in Wavelets, Multilevel Methods and Elliptic PDEs, M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, eds., Oxford University Press, New York, 1997, pp. 213--302.
46.
J. Xu and L. Zikatanov, Algebraic multigrid methods, Acta Numer., 26 (2017), pp. 591--721.
47.
Z. J. Zhang, W. H. Deng, and G. E. Karniadakis, A Riesz basis Galerkin method for the tempered fractional Laplacian, SIAM J. Numer. Anal., 56 (2018), pp. 3010--3039.
48.
A. Zoia, A. Rosso, and M. Kardar, Fractional Laplacian in bounded domains, Phys. Rev. E, 76 (2007), 021116.

## Information & Authors

### Information

#### Published In

SIAM Journal on Matrix Analysis and Applications
Pages: 1546 - 1570
ISSN (online): 1095-7162

#### History

Submitted: 29 August 2018
Accepted: 14 July 2020
Published online: 8 October 2020

### Authors

#### Affiliations

Stefano Serra-Capizzano

#### Funding Information

Fundamental Research Funds for the Central Universities https://doi.org/10.13039/501100012226 : lzujbky-2019-80
National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11601206