Methods and Algorithms for Scientific Computing

# Computing Fractional Laplacians on Complex-Geometry Domains: Algorithms and Simulations

## Abstract

We consider a fractional Laplacian defined in bounded domains by the eigen-decomposition of the integer-order Laplacian, and demonstrate how to compute very accurately (using the spectral element method) the eigenspectrum and corresponding eigenfunctions in two-dimensional prototype complex-geometry domains. We then employ these eigenfunctions as trial and test bases to first solve the fractional diffusion equation, and subsequently to simulate two-phase flow based on the Navier--Stokes equations combined with a fractional Allen--Cahn mass-preserving model. A key point to the effectiveness of an exponential convergence of this approach is the use of a weighted Gram--Schmidt orthonormalization of the eigenfunctions that guarantees accurate projection and recovery of spectral accuracy for smooth solutions. We demonstrate that even when only part of the eigenspectrum is computed accurately we can still obtain exponential convergence if we employ the complete set of the eigenvectors of the discrete Laplacian. Accuracy is also verified by computing the eigenfunctions on square, disk, and L-shaped domains and obtaining numerical solutions of the fractional diffusion equation for different fractional orders. This is accomplished without the need of solving any linear systems as the eigenfunction decomposition leads naturally to a system of ODEs, and hence no spatial discretization is employed during time stepping. In the second application of the method, we replace the integer-order Laplacian in the Allen--Cahn model with its fractional counterpart and a similar procedure is followed. However, for the Navier--Stokes equations we need to solve a linear system, which we invert using an efficient ADI scheme. We demonstrate the effectiveness of the fractional Navier--Stokes/Allen--Cahn solver for the rising bubble problem in a square domain, and compare the results with the integer-order system and also with results by a different treatment of the fractional diffusion model using one-dimensional fractional derivatives. The present model yields sharper interface thickness compared to the integer-order model for the same resolution while it preserves the isotropic diffusion, and hence it is a good candidate for phase-field modeling of multiphase fluid flows.

## References

1.
O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), pp. 368--379.
2.
O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), pp. 191--206.
3.
I. Babuška, B. Q. Guo, and E. P. Stephan, The h-p version of the boundary element method with geometric mesh on polygonal domains, Comput. Methods Appl. Mech. Engrg., 80 (1990), pp. 319--325.
4.
D. A. Benson, R. Schumer, M. M. Meerschaert, and S. W. Wheatcraft, Fractional dispersion, Lévy motion, and the MADE tracer tests, in Dispersion in Heterogeneous Geological Formations, Springer, Dordrecht, The Netherlands, 2002, pp. 211--240.
5.
D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), pp. 1403--1412.
6.
D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2006), pp. 1413--1423.
7.
C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, Vol. 5, 1997, North-Holland, Amsterdam, 1997, pp. 209--485.
8.
K. Burrage, N. Hale, and D. Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput., 34 (2012), pp. A2145--A2172.
9.
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), pp. 1245--1260.
10.
C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, 1988.
11.
A. V. Chechkin, R. Metzler, V. Y. Gonchar, J. Klafter, and L. V. Tanatarov, First passage and arrival time densities for Lévy flights and the failure of the method of images, J. Phys. A, 36 (2003), pp. L537--L544.
12.
L. Chen, J. Shen, and C. Xu, Spectral direction splitting scheme for incompressible Navier-Stokes equations, East Asian J. Appl. Math., 1 (2011), pp. 215--234.
13.
L. Chen, J. Shen, and C. Xu, A triangular spectral method for the Stokes equations, Numer. Math. Theory Methods Appl., 4 (2011), pp. 158--179.
14.
P. Constantin and M. Ignatova, Remarks on the fractional Laplacian with Dirichlet boundary conditions and applications, Int. Math. Res. Not. IMRN, 2017 (2017), pp. 1653--1673.
15.
D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.
16.
X. Feng, T. Tang, and J. Yang, Stabilized Crank-Nicolson/Adams-Bashforth schemes for phase field models, East Asian J. Appl. Math., 3 (2013), pp. 59--80.
17.
L. Fox, P. Henrici, and C. Moler, Approximations and bounds for eigenvalues of elliptic operators, SIAM J. Numer. Anal., 4 (1967), pp. 89--102.
18.
V. Gafiychuk, B. Datsko, and V. Meleshko, Mathematical modeling of time fractional reaction-diffusion systems, J. Math. Anal. Appl., 220 (2008), pp. 215--225.
19.
P. Gatto and J. S. Hesthaven, Numerical approximation of the fractional Laplacian via hp-finite elements, with an application to image denoising, J. Sci. Comput., 65 (2015), pp. 249--270.
20.
W. J. Gordon and C. A. Hall, Construction of curvilinear co-ordinate systems and applications to mesh generation, Internat. J. Numer. Methods Engrg., 7 (1973), pp. 461--477.
21.
W. J. Gordon and C. A. Hall, Transfinite element methods: Blending-function interpolation over arbitrary curved element domains, Numer. Math., 21 (1973), pp. 109--129.
22.
R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance III: The diffusion limit, Mathematical Finance, in Trends Math., Birkhäuser, Basel, 2001, pp. 171--180.
23.
M. Y. Hussaini and T. A. Zang, Spectral methods in fluid dynamics, Annu. Rev. Fluid Mech., 19 (1987), pp. 339--367.
24.
M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II)--with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9 (2006), pp. 333--349.
25.
G. E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, Oxford, 2013.
26.
R. C. Koeller, Applcation of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51 (1984), pp. 229--307.
27.
D. Kusnezov, A. Bulgac, and G. D. Dang, Quantum Lévy processes and fractional kinetics, Phys. Rev. Lett., 82 (1999), pp. 1136--1139.
28.
N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 180, Springer, Berlin, 1972.
29.
X. Liu and S. Oishi, Verified eigenvalue evaluation for the Laplacian over polygonal domains of arbitrary shape, SIAM J. Numer. Anal., 51 (2013), pp. 1634--1654.
30.
M. M. Meerschaert and E. Scalas, Coupled continuous time random walks in finance, Phys. A, 370 (2006), pp. 114--118.
31.
I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
32.
M. Raberto, E. Scalas, and F. Mainardi, Waiting-times and returns in high-frequency finanical data: An empirical study, Phys. A, 314 (2002), pp. 749--755.
33.
Z. Rong and C. Xu, Numerical approximation of acoustic waves by spectral element methods, Appl. Numer. Math., 58 (2008), pp. 999--1016.
34.
J. Shen, Efficient spectral-Galerkin methods III: Polar and cylindrical geometries, SIAM J. Sci. Comput., 18 (1997), pp. 1583--1604.
35.
J. Shen, Modeling and numerical approximation of two-phase incompressible flows by a phase-field approach, Multiscale Model. Anal. Mater. Simul., 22 (2011), pp. 147--195.
36.
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), pp. 67--112.
37.
F. Song and C. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, J. Comput. Phys., 299 (2015), pp. 196--214.
38.
F. Song, C. Xu, and G. E. Karniadakis, A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations, Comput. Methods Appl. Mech. Engrg., 305 (2016), pp. 376--404.
39.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Vol. 2, Princeton University Press, Princeton, NJ, 1970.
40.
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1995.
41.
Q. Yang, I. Turner, F. Liu, and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33 (2011), pp. 1159--1180.
42.
X. Yang, J. J. Feng, C. Liu, and J. Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys., 218 (2006), pp. 417--428.
43.
Z. Zhang, How many numerical eigenvalues can we trust?, J. Sci. Comput., 65 (2015), pp. 455--466.
44.
A. Zoia, A. Rosso, and M. Kardar, Fractional Laplacian in bounded domains, Phys. Rev. E (3), 76 (2007), 021116.

## Information & Authors

### Information

#### Published In

SIAM Journal on Scientific Computing
Pages: A1320 - A1344
ISSN (online): 1095-7197

#### History

Submitted: 2 June 2016
Accepted: 5 June 2016
Published online: 27 July 2017

### Authors

#### Funding Information

Office of the Secretary of Defense https://doi.org/10.13039/100005713 : W911NF-15-1-0562

#### Funding Information

National Natural Science Foundation of China https://doi.org/10.13039/501100001809 : 11471274