Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian

doi:10.1007/s42967-024-00368-z

Abstract

Abstract: A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems (BVPs) of the fractional Laplacian on arbitrary bounded domains. It was shown to have advantages of both finite difference (FD) and finite element methods, including their efficient implementation through the fast Fourier transform (FFT) and the ability to work for complex domains and with mesh adaptation. The purpose of this work is to study GoFD in a meshfree setting, a key to which is to construct the data transfer matrix from a given point cloud to a uniform grid. Two approaches are proposed, one based on the moving least squares fitting and the other based on the Delaunay triangulation and piecewise linear interpolation. Numerical results obtained for examples with convex and concave domains and various types of point clouds are presented. They show that both approaches lead to comparable results. Moreover, the resulting meshfree GoFD converges in a similar order as GoFD with unstructured meshes and finite element approximation as the number of points in the cloud increases. Furthermore, numerical results show that the method is robust to random perturbations in the location of the points.

Key words: Fractional Laplacian, Meshfree, Finite difference (FD), Arbitrary domain, Overlay grid

CLC Number:

65N06
35R11

Jinye Shen, Bowen Shi, Weizhang Huang. Meshfree Finite Difference Solution of Homogeneous Dirichlet Problems of the Fractional Laplacian[J]. Communications on Applied Mathematics and Computation, 2025, 7(2): 589-605.

TrendMD

References

1. Acosta, G., Bersetche, F.M., Borthagaray, J.P.: A short FE implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian. Comput. Math. Appl. 74, 784–816 (2017)
2. Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55, 472–495 (2017)
3. Ainsworth, M., Glusa, C.: Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver. Comput. Methods Appl. Mech. Eng. 327, 4–35 (2017)
4. Ainsworth, M., Glusa, C.: Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains. In: Contemporary Computational Mathematics—a Celebration of the 80th Birthday of Ian Sloan, pp. 17–57. Springer, Cham (2018)
5. Antil, H., Brown, T., Khatri, R., Onwunta, A., Verma, D., Warma, M.: Chapter 3 — Optimal control, numerics, and applications of fractional PDEs. In: Handbook of Numerical Analysis, vol. 23, pp. 87–114. Elsevier, Amsterdam (2022)
6. Antil, H., Dondl, P., Striet, L.: Approximation of integral fractional Laplacian and fractional PDEs via sinc-basis. SIAM J. Sci. Comput. 43, A2897–A2922 (2021)
7. Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of the integral fractional Laplacian. Numer. Math. 142, 235–278 (2019)
8. Burkardt, J., Wu, Y., Zhang, Y.: A unified meshfree pseudospectral method for solving both classical and fractional PDEs. SIAM J. Sci. Comput. 43, A1389–A1411 (2021)
9. Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4, 97–108 (1989). (Computational geometry (Waterloo, ON) (1987))
10. Du, N., Sun, H.-W., Wang, H.: A preconditioned fast finite difference scheme for space-fractional diffusion equations in convex domains. Comput. Appl. Math. 38, 14 (2019)
11. Du, Q., Ju, L., Lu, J.: A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems. Math. Comput. 88, 123–147 (2019)
12. Duo, S., van Wyk, H.W., Zhang, Y.: A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem. J. Comput. Phys. 355, 233–252 (2018)
13. Dyda, B., Kuznetsov, A., Kwaśnicki, M.: Fractional Laplace operator and Meijer G-function. Constr. Approx. 45, 427–448 (2017)
14. Faustmann, M., Karkulik, M., Melenk, J.M.: Local convergence of the FEM for the integral fractional Laplacian. SIAM J. Numer. Anal. 60, 1055–1082 (2022)
15. Hao, Z., Zhang, Z., Du, R.: Fractional centered difference scheme for high-dimensional integral fractional Laplacian. J. Comput. Phys. 424, 109851 (2021)
16. Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods. Applied Mathematical Sciences Series, vol. 174. Springer, New York (2011)
17. Huang, W., Shen, J.: A grid-overlay finite difference method for the fractional Laplacian on arbitrary bounded domains. SIAM J. Sci. Comput. (to appear). http:// arxiv. org/ abs/ arXiv: 2307. 14437 (2023)
18. Huang, Y., Oberman, A.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52, 3056–3084 (2014)
19. Huang, Y., Oberman, A.: Finite difference methods for fractional Laplacians. arXiv: 1611. 00164 (2016)
20. Li, H., Liu, R., Wang, L.-L.: Efficient Hermite spectral-Galerkin methods for nonlocal diffusion equations in unbounded domains. Numer. Math. Theory Methods Appl. 15, 1009–1040 (2022)
21. Lischke, A., Pang, G., Gulian, M., Song, F., Glusa, C., Zheng, X., Mao, Z., Cai, W., Meerschaert, M. M., Ainsworth, M., Karniadakis, G. E.: What is the fractional Laplacian? A comparative review with new results. J. Comput. Phys. 404, 109009 (2020)
22. Liu, G.R.: Meshfree Methods: Moving Beyond the Finite Element Method, 2nd edn. CRC Press, Boca Raton (2010)
23. Minden, V., Ying, L.: A simple solver for the fractional Laplacian in multiple dimensions. SIAM J. Sci. Comput. 42, A878–A900 (2020)
24. Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. 2006, 48391 (2006)
25. Ortigueira, M.D.: Fractional central differences and derivatives. J. Vib. Control 14, 1255–1266 (2008)
26. Pang, G., Chen, W., Fu, Z.: Space-fractional advection-dispersion equations by the Kansa method. J. Comput. Phys. 293, 280–296 (2015)
27. Pang, H.-K., Sun, H.-W.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012)
28. Shewchuk, J.R.: General-dimensional constrained Delaunay and constrained regular triangulations. I. Combinatorial properties. Discrete Comput. Geom. 39, 580–637 (2008)
29. Somasekhar, M., Vivek, S., Malagi, K.S., Ramesh, V., Deshpande, S.M.: Adaptive cloud refinement (ACR)-adaptation in meshless framework. Commun. Comput. Phys. 11, 1372–1385 (2012)
30. Song, F., Xu, C., Karniadakis, G.E.: Computing fractional Laplacians on complex-geometry domains: algorithms and simulations. SIAM J. Sci. Comput. 39, A1320–A1344 (2017)
31. Suchde, P., Jacquemin, T., Davydov, O.: Point cloud generation for meshfree methods: an overview. Arch. Comput. Methods Eng. 30, 889–915 (2023)
32. Sun, J., Nie, D., Deng, W.: Algorithm implementation and numerical analysis for the two-dimensional tempered fractional Laplacian. BIT 61, 1421–1452 (2021)
33. Tian, X., Du, Q.: Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51, 3458–3482 (2013)
34. Trobec, R., Kosec, G.: Parallel Scientific Computing: Theory, Algorithms, and Applications of Mesh Based and Meshless Methods. SpringerBriefs in Computer Science, Springer, Cham (2015)
35. Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34, A2444–A2458 (2012)

[1]	Joseph Hunter, Zheng Sun, Yulong Xing. Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation [J]. Communications on Applied Mathematics and Computation, 2024, 6(1): 658-687.
[2]	Yu Wang, Min Cai. Finite Difference Schemes for Time-Space Fractional Diffusion Equations in One- and Two-Dimensions [J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1674-1696.
[3]	Yaoqiang Yan, Weihua Deng, Daxin Nie. A Finite-Diference Approximation for the One- and Two-Dimensional Tempered Fractional Laplacian [J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 129-145.
[4]	Zhiping Mao, Zhen Li, George Em Karniadakis. Nonlocal Flocking Dynamics: Learning the Fractional Order of PDEs from Particle Simulations [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 597-619.
[5]	Kamran Kazmi, Abdul Khaliq. A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions [J]. Communications on Applied Mathematics and Computation, 2019, 1(4): 525-544.