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Numerical methods for nonlocal and fractional models

Published online by Cambridge University Press:  30 November 2020

Marta D’Elia ,
Qiang Du ,
Christian Glusa ,
Max Gunzburger ,
Xiaochuan Tian  and
Zhi Zhou
Marta D’Elia
Affiliation:
Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico87185, USA E-mail: [email protected], [email protected]://sites.google.com/site/martadeliawebsite
Qiang Du
Affiliation:
Department of Applied Physics and Applied Mathematics and Data Science Institute, Columbia University, New York, NY10027, USA E-mail: [email protected]://www.columbia.edu/~qd2125
Christian Glusa
Affiliation:
Center for Computing Research, Sandia National Laboratories, Albuquerque, New Mexico87185, USA E-mail: [email protected], [email protected]://sites.google.com/site/martadeliawebsite
Max Gunzburger
Affiliation:
Department of Scientific Computing, Florida State University, Tallahassee, Florida32306, USA E-mail: [email protected]://www.sc.fsu.edu/gunzburg
Xiaochuan Tian
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas78751, USA E-mail: [email protected]://web.ma.utexas.edu/users/xtian
Zhi Zhou
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China E-mail: [email protected]://sites.google.com/site/zhizhou0125
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Abstract

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

Type
Research Article
Information
Acta Numerica , Volume 29 , May 2020 , pp. 1 - 124
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES 8

Abatangelo, N. and Dupaigne, L. (2017), ‘Nonhomogeneous boundary conditions for the spectral fractional Laplacian’, Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 439–467.Google Scholar
Abramowitz, M. and Stegun, I. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards and Dover.Google Scholar
Acosta, G. and Borthagaray, J.-P. (2017), ‘A fractional Laplace equation: Regularity of solutions and finite element approximations’, SIAM J. Numer. Anal. 55, 472–495.10.1137/15M1033952CrossRefGoogle Scholar
Acosta, G., Bersetche, F. and Borthagaray, J. (2017), ‘A short FE implementation for a 2D homogeneous Dirichlet problem of a fractional Laplacian’, Comput. Math. Appl. 74, 784–816.10.1016/j.camwa.2017.05.026CrossRefGoogle Scholar
Acosta, G., Borthagaray, J.-P. and Heuer, N. (2019), ‘Finite element approximations of the nonhomogeneous fractional Dirichlet problem’, IMA J. Numer. Anal. 39, 1471–1501.Google Scholar
Acosta, G., Borthagaray, J.-P., Bruno, O. and Maas, M. (2018), ‘Regularity theory and high order numerical methods for the (1D)-fractional Laplacian’, Math. Comp. 87, 1821–1857.10.1090/mcom/3276CrossRefGoogle Scholar
Ainsworth, M. and Glusa, C. (2017), ‘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. Engrg 327, 4–35.10.1016/j.cma.2017.08.019CrossRefGoogle Scholar
Ainsworth, M. and Glusa, C. (2018 a), ‘Hybrid finite element-spectral method for the fractional Laplacian approximation theory and efficient solver’, SIAM J. Sci. Comput. 40, A2383–A2405.Google Scholar
Ainsworth, M. and Glusa, C. (2018 b), 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 (Dick, J. et al., eds), Springer, pp. 17–57.10.1007/978-3-319-72456-0_2CrossRefGoogle Scholar
Ainsworth, M. and Mao, Z. (2017), ‘Analysis and approximation of a fractional Cahn–Hilliard equation’, SIAM J. Numer. Anal. 55, 1689–1718.10.1137/16M1075302CrossRefGoogle Scholar
Ainsworth, M. and McLean, W. (2003), ‘Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes’, Numer. Math. 93, 387–413.Google Scholar
Ainsworth, M., McLean, W. and Tran, T. (1999), ‘The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling’, SIAM J. Numer. Anal. 36, 1901–1932.Google Scholar
Aksoylu, B. and Mengesha, T. (2010), ‘Results on nonlocal boundary value problems’, Numer. Funct. Anal. Optim. 31, 1301–1317.CrossRefGoogle Scholar
Aksoylu, B. and Parks, M. (2011), ‘Variational theory and domain decomposition for nonlocal problems’, Appl. Math. Comput. 217, 6498–6515.Google Scholar
Aksoylu, B. and Unlu, Z. (2014), ‘Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces’, SIAM J. Numer. Anal. 52, 653–677.Google Scholar
Alali, B., Gunzburger, M. and Liu, K. (2015), ‘A generalized nonlocal calculus’, J. Appl. Math. Phys. 223, 2807–2828.Google Scholar
Andreu, F., Mazón, J., Rossi, J. and Toledo, J. (2010), Nonlocal Diffusion Problems, Vol. 165 of Mathematical Surveys and Monographs , American Mathematical Society.Google Scholar
Antil, H. and Bartels, S. (2017), ‘Spectral approximation of fractional PDEs in image processing and phase field modeling’, Comput. Methods Appl. Math. 17, 661–678.Google Scholar
Antil, H. and Warma, M. (2020), ‘Optimal control of fractional semilinear PDEs’, ESAIM Control Optim. Calc. Var. 26, 5.10.1051/cocv/2019003CrossRefGoogle Scholar
Antil, H., Khatri, R. and Warma, M. (2019 a), ‘External optimal control of nonlocal PDEs’, Inverse Problems 35, 084003.Google Scholar
Antil, H., Otarola, E. and Salgado, A. J. (2016), ‘A space-time fractional optimal control problem: Analysis and discretization’, SIAM J. Control Optim. 54, 1295–1328.10.1137/15M1014991CrossRefGoogle Scholar
Antil, H., Otarola, E. and Salgado, A. J. (2018), ‘Optimization with respect to order in a fractional diffusion model: Analysis, approximation and algorithmic aspects’, J. Sci. Comput. 77, 204–224.10.1007/s10915-018-0703-0CrossRefGoogle Scholar
Antil, H., Pfefferer, J. and Rogovs, S. (2018), ‘Fractional operators with inhomogeneous boundary conditions: Analysis, control, and discretization’, Commun. Math. Sci. 16, 1395–1426.Google Scholar
Antil, H., Verma, D. and Warma, M. (2019 b), External optimal control of fractional parabolic PDEs. arXiv:1904.07123 Google Scholar
Atkinson, K. and Han, W. (2012), Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Vol. 2044 of Lecture Notes in Mathematics, Springer.10.1007/978-3-642-25983-8CrossRefGoogle Scholar
Babuška, I. (1971), ‘Error-bounds for finite element method’, Numer. Math. 16, 322–333.10.1007/BF02165003CrossRefGoogle Scholar
Babuška, I. and Aziz, K. (1972), Survey lectures on the mathematical foundation of the finite element method. In The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations, Academic Press, pp. 1–359.Google Scholar
Baeumer, B. and Meerschaert, M. (2010), ‘Tempered stable Lévy motion and transient super-diffusion’, J. Comput. Appl. Math. 233, 2438–2448.10.1016/j.cam.2009.10.027CrossRefGoogle Scholar
Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. (2016), Fractional Calculus: Models and Numerical Methods, World Scientific.10.1142/10044CrossRefGoogle Scholar
Banjai, L., Melenk, J., Nochetto, R., Otárola, E., Salgado, A. and Schwab, C. (2019), ‘Tensor FEM for spectral fractional diffusion’, Found. Comput. Math. 19, 901–962.10.1007/s10208-018-9402-3CrossRefGoogle Scholar
Biccari, U. and Hernández-Santamaría, V. (2018), ‘The Poisson equation from non-local to local’, Electron. J. Differential Equations 2018, 145.Google Scholar
Blumenthal, R., Getoor, R. and Ray, D. (1961), ‘On the distribution of first hits for the symmetric stable processes’, Trans. Amer. Math. Soc. 99, 540–554.Google Scholar
Bobaru, F. and Duangpanya, M. (2010), ‘The peridynamic formulation for transient heat conduction’, Int. J. Heat Mass Transfer 53, 4047–4059.CrossRefGoogle Scholar
Bobaru, F. and Zhang, G. (2015), ‘Why do cracks branch? A peridynamic investigation of dynamic brittle fracture’, Int. J. Fracture 196, 59–98.10.1007/s10704-015-0056-8CrossRefGoogle Scholar
Bobaru, F., Yang, M., Alves, L., Silling, S., Askari, E. and Xu, J. (2009), ‘Convergence, adaptive refinement, and scaling in 1D peridynamics’, Inter. J. Numer. Meth. in Eng. 77, 852–877.CrossRefGoogle Scholar
Bogdan, K. and Byczkowski, T. (1999), ‘Potential theory for the $\alpha$ -stable Schrödinger operator on bounded Lipschitz domains’, Studia Math. 133, 53–92.CrossRefGoogle Scholar
Bogdan, K., Burdzy, K. and Chen, Z.-Q. (2003), ‘Censored stable processes’, Probab. Theory Related Fields 127, 89–152.10.1007/s00440-003-0275-1CrossRefGoogle Scholar
Bonito, A. and Pasciak, J. (2015), ‘Numerical approximation of fractional powers of elliptic operators’, Math. Comp. 84, 2083–2110.10.1090/S0025-5718-2015-02937-8CrossRefGoogle Scholar
Bonito, A. and Pasciak, J. (2017), ‘Numerical approximation of fractional powers of regularly accretive operators’, IMA J. Numer. Anal. 37, 1245–1273.CrossRefGoogle Scholar
Bonito, A., Lei, W. and Pasciak, J. (2019), ‘Numerical approximation of the integral fractional Laplacian’, Numer. Math. 142, 235–278.Google Scholar
Bonito, A., Lei, W. and Salgado, A. (2020), ‘Finite element approximation of an obstacle problem for a class of integro-differential operators’, ESAIM Math. Model. Numer. Anal. 54, 229–253.CrossRefGoogle Scholar
Borthagaray, J.-P., Nochetto, R. and Salgado, A. (2019), ‘Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian’, Math. Model. Meth. Appl. Sci. 29, 2670–2717.10.1142/S021820251950057XCrossRefGoogle Scholar
Bourgain, J., Brezis, H. and Mironescu, P. (2001), Another Look at Sobolev Spaces, IOS Press.Google Scholar
Bourgoin, M., Ouellette, N., Xu, H., Berg, J. and Bodenschatz, E. (2006), ‘The role of pair dispersion in turbulent flow’, Science 311, 835–838.10.1126/science.1121726CrossRefGoogle ScholarPubMed
Boyd, J. (1987), ‘Spectral methods using rational basis functions on an infinite interval’, J. Comput. Phys. 69, 112–142.CrossRefGoogle Scholar
Boyd, J. (2001), ‘Rational Chebyshev spectral methods for unbounded solutions on an infinite interval using polynomial-growth special basis functions’, Comput. Math. Appl. 41, 1293–1315.CrossRefGoogle Scholar
Brenner, S. and Scott, R. (1994), The Mathematical Theory of Finite Element Methods, Springer.Google Scholar
Buades, A., Coll, B. and Morel, J. (2008), ‘Nonlocal image and movie denoising’, Int. J. Comput. Vision 76, 123–139.CrossRefGoogle Scholar
Buades, A., Coll, B. and Morel, J. (2010), ‘Image denoising methods: A new nonlocal principle’, SIAM Rev. 52, 113–147.10.1137/090773908CrossRefGoogle Scholar
Bucur, C. (2016), ‘Some observations on the Green function for the ball in the fractional Laplace framework’, Commun. Pure Appl. Anal. 15, 657–699.CrossRefGoogle Scholar
Buhmann, M. (2000), Radial basis functions. In Acta Numerica , Vol. 9, Cambridge University Press, pp. 1–38.Google Scholar
Buhmann, M. (2003), Radial Basis Functions: Theory and Implementations, Vol. 12 of Cambridge Monographs on Applied and Computational Mathematics , Cambridge University Press.Google Scholar
Burch, N. and Lehoucq, R. (2015), ‘Computing the exit-time for a finite-range symmetric jump process’, Monte Carlo Methods Appl. 21, 139–152.10.1515/mcma-2014-0015CrossRefGoogle Scholar
Burch, N., D’Elia, M. and Lehoucq, R. (2014), ‘The exit-time problem for a Markov jump process’, Eur. Phys. J. Special Topics 223, 3257–3271.CrossRefGoogle Scholar
Burkovska, O. and Gunzburger, M. (2019 a), Affine approximation of parametrized kernels and model order reduction for nonlocal and fractional Laplace models. arXiv:1901.06748 Google Scholar
Burkovska, O. and Gunzburger, M. (2019 b), ‘Regularity and approximation analyses of nonlocal variational equality and inequality problems’, J. Math. Anal. Appl. 478, 1027–1048.10.1016/j.jmaa.2019.05.064CrossRefGoogle Scholar
Caffarelli, L. and Silvestre, L. (2007), ‘An extension problem related to the fractional Laplacian’, Comm. Partial Differential Equations 32, 1245–1260.CrossRefGoogle Scholar
Caffarelli, L. and Stinga, P. (2016), ‘Fractional elliptic equations, Caccioppoli estimates and regularity’, Ann. Inst. H. Poincaré Anal. Non Linéaire 33, 767–807.Google Scholar
Caffarelli, L., Ros-Oton, X. and Serra, J. (2017), ‘Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries’, Invent. Math. 208, 1155–1211.CrossRefGoogle Scholar
Çelik, C. and Duman, M. (2012), ‘Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative’, J. Comput. Phys. 231, 1743–1750.10.1016/j.jcp.2011.11.008CrossRefGoogle Scholar
Chen, A., Du, Q., Li, C. and Zhou, Z. (2017), ‘Asymptotically compatible schemes for space-time nonlocal diffusion equations’, Chaos Solitons Fractals 102, 361–371.CrossRefGoogle Scholar
Chen, L., Nochetto, R., Otárola, E. and Salgado, A. J. (2016a), ‘Multilevel methods for nonuniformly elliptic operators and fractional diffusion’, Math. Comp. 85, 2583–2607.CrossRefGoogle Scholar
Chen, M. and Deng, W. (2014), ‘Fourth order accurate scheme for the space fractional diffusion equations’, SIAM J. Numer. Anal. 52, 1418–1438.10.1137/130933447CrossRefGoogle Scholar
Chen, S., Shen, J. and Wang, L.-L. (2016b), ‘Generalized Jacobi functions and their applications to fractional differential equations’, Math. Comp. 85, 1603–1638.CrossRefGoogle Scholar
Chen, W. (2006), ‘A speculative study of 2/3-order fractional Laplacian modeling of turbulence: Some thoughts and conjectures’, Chaos 16, 023126.10.1063/1.2208452CrossRefGoogle Scholar
Chen, X. and Gunzburger, M. (2011), ‘Continuous and discontinuous finite element methods for a peridynamics model of mechanics’, Comput. Methods Appl. Mech. Engrg 200, 1237–1250.CrossRefGoogle Scholar
Chen, Z.-Q. and Kim, P. (2002), ‘Green function estimate for censored stable processes’, Probab. Theory Related Fields 124, 595–610.CrossRefGoogle Scholar
Chernov, A., von Petersdorff, T. and Schwab, C. (2011), ‘Exponential convergence of ${hp}$ quadrature for integral operators with Gevrey kernels’, ESAIM Math. Model. Numer. Anal. 45, 387–422.CrossRefGoogle Scholar
Ciarlet, P. (2002), Finite Element Methods for Elliptic Problems, SIAM.CrossRefGoogle Scholar
Cortazar, C., Elgueta, M., Rossi, J. and Wolanski, N. (2008), ‘How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems’, Arch. Rat. Mech. Anal. 187, 137–156.CrossRefGoogle Scholar
Coulhon, T. and Grigoryan, A. (1998), ‘Random walks on graphs with regular volume growth’, Geom. Funct. Anal. 8, 656–701.CrossRefGoogle Scholar
Curtiss, J. (1950), Sampling methods applied to differential and difference equations. In Proceedings, Seminar on Scientific Computation, November 1949, IBM , pp. 87–109.Google Scholar
Cusimano, N., del Teso, F., Gerardo-Giorda, L. and Pagnini, G. (2018), ‘Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions’, SIAM J. Numer. Anal. 56, 1243–1272.10.1137/17M1128010CrossRefGoogle Scholar
Darbon, J., Cunha, A., Chan, T., Osher, S. and Jensen, G. (2008), Fast nonlocal filtering applied to electron cryomicroscopy. In 2008 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro, IEEE, pp. 1331–1334.10.1109/ISBI.2008.4541250CrossRefGoogle Scholar
Del Teso, F., Endal, J. and Jakobsen, E. (2018), ‘Robust numerical methods for nonlocal (and local) equations of porous medium type, II: Schemes and experiments’, SIAM J. Numer. Anal. 56, 3611–3647.CrossRefGoogle Scholar
Del Teso, F., Endal, J. and Jakobsen, E. (2019), ‘Robust numerical methods for nonlocal (and local) equations of porous medium type, I: Theory’, SIAM J. Numer. Anal. 57, 2266–2299.CrossRefGoogle Scholar
D’Elia, M. and Gunzburger, M. (2013), ‘The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator’, Comput. Math. Appl. 66, 1245–1260.CrossRefGoogle Scholar
D’Elia, M. and Gunzburger, M. (2014), ‘Optimal distributed control of nonlocal steady diffusion problems’, SIAM J. Control Optim. 55, 667–696.Google Scholar
D’Elia, M. and Gunzburger, M. (2016), ‘Identification of the diffusion parameter in nonlocal steady diffusion problems’, Appl. Math. Optim. 73, 227–249.Google Scholar
D’Elia, M., De los Reyes, J.-C. and Trujillo, A. M. (2019 c), Bilevel parameter optimization for nonlocal image denoising models. arXiv:1912.02347 CrossRefGoogle Scholar
D’Elia, M., Du, Q., Gunzburger, M. and Lehoucq, R. B. (2017), ‘Nonlocal convection–diffusion problems on bounded domains and finite-range jump processes’, Comput. Methods Appl. Math. 17, 707–722.CrossRefGoogle Scholar
D’Elia, M., Flores, C., Li, X., Radu, P. and Yu, Y. (2019 a), Helmholtz–Hodge decompositions in the nonlocal framework: Well-posedness analysis and applications. arXiv:1908.08624 CrossRefGoogle Scholar
D’Elia, M., Glusa, C. and Otárola, E. (2019 b), ‘A priori error estimates for the optimal control of the integral fractional Laplacian’, SIAM J. Control Optim. 57, 2775–2798.Google Scholar
D’Elia, M., Tian, X. and Yu, Y. (2019 d), A physically-consistent, flexible and efficient strategy to convert local boundary conditions into nonlocal volume constraints. arXiv:1906.04259 Google Scholar
Deng, B., Zhang, Z. and Zhao, X. (2019), ‘Superconvergence points for the spectral interpolation of Riesz fractional derivatives’, J. Sci. Comput. 81, 1577–1601.Google Scholar
Deng, W., Li, B., Tian, W., and Zhang, P., (2018), ‘Boundary problems for the fractional and tempered fractional operators’, Multiscale Model. Simul. 16, 125–149.CrossRefGoogle Scholar
Di Nezza, E., Palatucci, G. and Valdinoci, E. (2012), ‘Hitchhiker’s guide to the fractional Sobolev spaces’, Bull. Sci. Math. 136, 521–573.Google Scholar
Ding, H., Li, C. and Chen, Y.-Q. (2015), ‘High-order algorithms for Riesz derivative and their applications, II’, J. Comput. Phys. 293, 218–237.Google Scholar
Dipierro, S., Ros-Oton, X. and Valdinoci, E. (2017), ‘Nonlocal problems with Neumann boundary conditions’, Rev. Mat. Iberoamer. 33, 377–416.Google Scholar
Du, Q. (2019), Nonlocal Modeling, Analysis, and Computation, Vol. 94 of CBMS-NSF Conference Series in Applied Mathematics, SIAM.Google Scholar
Du, Q. and Feng, X. (2020), The phase field method for geometric moving interfaces and their numerical approximations. In Geometric Partial Differential Equations , Part I, Vol. 21 of Handbook of Numerical Analysis, Elsevier.Google Scholar
Du, Q. and Huang, Z. (2017), ‘Numerical solution of a scalar one-dimensional monotonicity-preserving nonlocal conservation law’, J. Math. Res. Appl. 37, 1–18.Google Scholar
Du, Q. and Tian, X. (2014), Asymptotically compatible schemes for peridynamics based on numerical quadratures. In ASME 2014 International Mechanical Engineering Congress and Exposition, IMECE2014–39620.CrossRefGoogle Scholar
Du, Q. and Tian, X. (2015), Robust discretization of nonlocal models related to peridynamics. In Meshfree Methods for Partial Differential Equations VII , Vol. 100 of Lecture Notes in Computational Science and Engineering, Springer, pp. 97–113.CrossRefGoogle Scholar
Du, Q. and Tian, X. (2018), Heterogeneously localized nonlocal operators, boundary traces and variational problems. In Proceedings of the International Congress of Chinese Mathematicians, Vol. 43 of the ALM Series, International Press, pp. 217–236.Google Scholar
Du, Q. and Tian, X. (2019), ‘Mathematics of smoothed particle hydrodynamics via a nonlocal Stokes equation’, Found. Comput. Math. doi: 10.1007/s10208-019-09432-0 CrossRefGoogle Scholar
Du, Q. and Yang, J. (2016), ‘Asymptotically compatible Fourier spectral approximations of nonlocal Allen–Cahn equations’, SIAM J. Numer. Anal. 54, 1899–1919.CrossRefGoogle Scholar
Du, Q. and Yang, J. (2017), ‘Fast and accurate implementation of Fourier spectral approximations of nonlocal diffusion operators and its applications’, J. Comput. Phys. 332, 118–134.10.1016/j.jcp.2016.11.028CrossRefGoogle Scholar
Du, Q. and Yin, X. (2019), ‘A conforming DG method for linear nonlocal models with integrable kernels’, J. Sci. Comput. 80, 1913–1935.CrossRefGoogle Scholar
Du, Q. and Zhou, K. (2011), ‘Mathematical analysis for the peridynamic nonlocal continuum theory’, ESAIM Math. Model. Numer. Anal. 45, 217–234.10.1051/m2an/2010040CrossRefGoogle Scholar
Du, Q. and Zhou, Z. (2017), Multigrid finite element method for nonlocal diffusion equations with a fractional kernel. Preprint.Google Scholar
Du, Q., Gunzburger, M., Lehoucq, R. and Zhou, K. (2012 a), ‘Analysis and approximation of nonlocal diffusion problems with volume constraints’, SIAM Rev. 54, 667–696.10.1137/110833294CrossRefGoogle Scholar
Du, Q., Gunzburger, M., Lehoucq, R. and Zhou, K. (2013 a), ‘A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws’, Math. Model. Meth. Appl. Sci. 23, 493–540.CrossRefGoogle Scholar
Du, Q., Han, H., Zheng, C. and Zhang, J. (2018 a), ‘Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains’, SIAM J. Sci. Comput. 40, A1430–1445.CrossRefGoogle Scholar
Du, Q., Huang, Z. and LeFloch, P. (2017 a), ‘Nonlocal conservation laws, I: A new class of monotonicity-preserving models’, SIAM J. Numer. Anal. 55, 2465–2489.CrossRefGoogle Scholar
Du, Q., Huang, Z. and Lehoucq, R. (2014), ‘Nonlocal convection–diffusion volume-constrained problems and jump processes’, Discrete Contin. Dyn. Syst. B 19, 961–977.Google Scholar
Du, Q., Ju, L. and Lu, J. (2019 a), ‘Analysis of fully discrete approximations for dissipative systems and application to time-dependent nonlocal diffusion problems’, J. Sci. Comput. 78, 1438–1466.CrossRefGoogle Scholar
Du, Q., Ju, L. and Lu, J. (2019 b), ‘A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems’, Math. Comp. 88, 123–147.10.1090/mcom/3333CrossRefGoogle Scholar
Du, Q., Ju, L., Li, X. and Qiao, Z. (2018 b), ‘Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation’, J. Comput. Phys. 363, 39–54.CrossRefGoogle Scholar
Du, Q., Ju, L., Lu, J. and Tian, X. (2020 a), ‘A discontinuous Galerkin method with penalty for one-dimensional nonlocal diffusion problems’, Comm. Appl. Math. Comput. 2, 31–55.CrossRefGoogle Scholar
Du, Q., Ju, L., Tian, L. and Zhou, K. (2013 b), ‘A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models’, Math. Comp. 82, 1889–1922.CrossRefGoogle Scholar
Du, Q., Kamm, J., Lehoucq, R. and Parks, M. (2012 b), ‘A new approach for a nonlocal, nonlinear conservation law’, SIAM J. Appl. Math. 72, 464–487.CrossRefGoogle Scholar
Du, Q., Lipton, R. and Mengesha, T. (2016 a), ‘Multiscale analysis of linear evolution equations with applications to nonlocal models for heterogeneous media’, ESAIM Math. Model. Numer. Anal. 50, 1425–1455.CrossRefGoogle Scholar
Du, Q., Tao, Y. and Tian, X. (2018 c), ‘A peridynamic model of fracture mechanics with bond-breaking’, J. Elasticity 132, 197–218.10.1007/s10659-017-9661-2CrossRefGoogle Scholar
Du, Q., Tao, Y., Tian, X. and Yang, J. (2016 b), ‘Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients’, Comput. Methods Appl. Mech. Engrg 310, 605–627.CrossRefGoogle Scholar
Du, Q., Tao, Y., Tian, X. and Yang, J. (2019 c), ‘Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximations of nonlocal Green’s functions’, IMA J. Numer. Anal. 39, 607–625.Google Scholar
Du, Q., Tian, L. and Zhao, X. (2013 c), ‘A convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models’, SIAM J. Numer. Anal. 51, 1211–1234.10.1137/120871638CrossRefGoogle Scholar
Du, Q., Toniazzi, L. and Zhou, Z. (2020 b), ‘Stochastic representation of solution to nonlocal-in-time diffusion’, Stoch. Proc. Appl. 10, 2058–2085.10.1016/j.spa.2019.06.011CrossRefGoogle Scholar
Du, Q., Yang, J. and Zhou, Z. (2017 b), ‘Analysis of a nonlocal-in-time parabolic equation’, Discrete Contin. Dyn. Syst. B 22, 339–368.Google Scholar
Du, Q., Zhang, J. and Zheng, C. (2018 d), ‘Nonlocal wave propagation in unbounded multi-scale media’, Commun. Comput. Phys. 24, 1049–1072.CrossRefGoogle Scholar
Duo, S. and Zhang, Y. (2019), ‘Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications’, Comput. Methods Appl. Mech. Engrg 355, 639–662.CrossRefGoogle Scholar
Duo, S., van Wyk, H.-W. and Zhang, Y. (2018), ‘A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem’, J. Comput. Phys. 355, 233–252.CrossRefGoogle Scholar
Dyda, B., Kuznetsov, A. and Kwaśnicki, M. (2017 a), ‘Eigenvalues of the fractional Laplace operator in the unit ball’, J. Lond. Math. Soc. ( 2 ) 95, 500–518.CrossRefGoogle Scholar
Dyda, B., Kuznetsov, A. and Kwaśnicki, M. (2017 b), ‘Fractional Laplace operator and Meijer G-function’, Constr. Approx. 45, 427–448.CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2004), Theory and Practice of Finite Elements, Springer.10.1007/978-1-4757-4355-5CrossRefGoogle Scholar
Ervin, V. and Roop, J. (2007), ‘Variational solution of fractional advection dispersion equations on bounded domains in ${\mathbb{R}}^d$ ’, Numer. Methods Partial Differential Equations 23, 256–281.Google Scholar
Ervin, V., Heuer, N. and Roop, J. (2018), ‘Regularity of the solution to 1-D fractional order diffusion equations’, Math. Comp. 87, 2273–2294.CrossRefGoogle Scholar
Evans, L. (1998), Partial Differential Equations, American Mathematical Society.Google Scholar
Faustmann, M., Melenk, J., Parvizi, M. and Praetorius, D. (2019), Optimal adaptivity and preconditioning for the fractional Laplacian. In Proceedings of WONAPDE.Google Scholar
Felsinger, M., Kassmann, M. and Voigt, P. (2015), ‘The Dirichlet problem for nonlocal operators’, Math. Z. 279, 779–809.CrossRefGoogle Scholar
Fuensanta, A. and Muñoz, J. (2015), ‘Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design’, J. Math. Anal. Appl. 429, 288–310.Google Scholar
Gal, C. and Warma, M. (2016), ‘Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions’, Evol. Equ. Control Theory 5, 61–103.Google Scholar
Gilboa, G. and Osher, S. (2007), ‘Nonlocal linear image regularization and supervised segmentation’, Multiscale Model. Simul. 6, 595–630.CrossRefGoogle Scholar
Gilboa, G. and Osher, S. (2008), ‘Nonlocal operators with applications to image processing’, Multiscale Model. Simul. 7, 1005–1028.CrossRefGoogle Scholar
Girault, V. and Raviart, P.-A. (1986), Finite Element Methods for Navier–Stokes Equations, Springer.CrossRefGoogle Scholar
Glusa, C. and Otárola, E. (2019), Optimal control of a parabolic fractional PDE: Analysis and discretization. arXiv:1905.10002 Google Scholar
Gradshteyn, I. and Ryzhik, I. (2007), Table of Integrals, Series, and Products, Elsevier/Academic Press.Google Scholar
Grubb, G. (2015 a), ‘Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$ -transmission pseudodifferential operators’, Adv. Math. 268, 478–528.CrossRefGoogle Scholar
Grubb, G. (2015 b), ‘Spectral results for mixed problems and fractional elliptic operators’, J. Math. Anal. Appl. 421, 1616–1634.CrossRefGoogle Scholar
Grubb, G. (2016), ‘Regularity of spectral fractional Dirichlet and Neumann problems’, Math. Nachrichten 289, 831–844.Google Scholar
Guan, Q. and Gunzburger, M. (2015), ‘Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation’, Numer. Method. Partial Differential Equations 31, 500–516.CrossRefGoogle Scholar
Guan, Q. and Gunzburger, M. (2017), ‘Analysis and approximation of a nonlocal obstacle problem’, J. Comput. Appl. Math. 313, 102–118.CrossRefGoogle Scholar
Guan, Q., Gunzburger, M., Webster, C. and Zhang, G. (2017), ‘Reduced basis methods for nonlocal diffusion problems with random input data’, Comput. Methods Appl. Mech. Engrg 317, 746–770.CrossRefGoogle Scholar
Gunzburger, M. and Lehoucq, R. (2010), ‘A nonlocal vector calculus with application to nonlocal boundary value problems’, Multiscale Model. Simul. 8, 1581–1598.CrossRefGoogle Scholar
Gunzburger, M., Jiang, N. and Xu, F. (2018), ‘Analysis and approximation of a fractional Laplacian-based closure model for turbulent flows and its connection to Richardson pair dispersion’, Comput. Math. Appl. 75, 1973–2001.CrossRefGoogle Scholar
Guo, B. and Wang, L. (2004), ‘Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces’, J. Approx. Theory 128, 1–41.Google Scholar
Hackbusch, W. (1985), Multi-Grid Methods and Applications, Springer.CrossRefGoogle Scholar
Hackbusch, W. (1994), Iterative Solution of Large Sparse Systems of Equations, Springer.CrossRefGoogle Scholar
Hu, W., Ha, Y. and Bobaru, F. (2012), ‘Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites’, Comput. Methods Appl. Mech. Engrg 217, 247–261.10.1016/j.cma.2012.01.016CrossRefGoogle Scholar
Huang, Y. and Oberman, A. (2014), ‘Numerical methods for the fractional Laplacian: A finite difference-quadrature approach’, SIAM J. Numer. Anal. 52, 3056–3084.CrossRefGoogle Scholar
Jha, P. and Lipton, R. (2019), ‘Numerical convergence of finite difference approximations for state based peridynamic fracture models’, Comput. Methods Appl. Mech. Engrg 351, 184–225.CrossRefGoogle Scholar
Jha, P. and Lipton, R. (2020), ‘Finite element convergence for state-based peridynamic fracture models’, Comm. Appl. Math. Comput. 2, 93–128.CrossRefGoogle Scholar
Jiang, X., Lim, L.-H., Yao, Y. and Ye, Y. (2011), ‘Statistical ranking and combinatorial Hodge theory’, Math. Prog. 127, 203–244.Google Scholar
Kac, M. (1949), ‘On distributions of certain Wiener functionals’, Trans. Amer. Math. Soc. 65, 1–13.Google Scholar
Karniadakis, G., ed. (2019), Handbook of Fractional Calculus with Applications, Vol. 3, Numerical Methods, De Gruyter.Google Scholar
Kindermann, S., Osher, S. and Jones, P. (2005), ‘Deblurring and denoising of images by nonlocal functionals’, SIAM Multiscale Model. Simul. 4, 1091–1115.CrossRefGoogle Scholar
Kyprianou, A., Osojnik, A. and Shardlow, T. (2018), ‘Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian’, IMA J. Numer. Anal. 38, 1550–1578.CrossRefGoogle Scholar
Lee, H. and Du, Q. (2019), ‘Asymptotically compatible SPH-like particle discretizations of one dimensional linear advection models’, SIAM J. Numer. Anal. 57, 127–147.CrossRefGoogle Scholar
Lee, H. and Du, Q. (2020), ‘Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications’, ESAIM Math. Model. Numer. Anal. 54, 105–128.CrossRefGoogle Scholar
Lehoucq, R. and Rowe, S. (2016), ‘A radial basis function Galerkin method for inhomogeneous nonlocal diffusion’, Comput. Methods Appl. Mech. Engrg 299, 366–380.Google Scholar
Lehoucq, R., Narcowich, F., Rowe, S. and Ward, J. (2018), ‘A meshless Galerkin method for non-local diffusion using localized kernel bases’, Math. Comp. 87, 2233–2258.CrossRefGoogle Scholar
Lei, S.-L. and Sun, H.-W. (2013), ‘A circulant preconditioner for fractional diffusion equations’, J. Comput. Phys. 242, 715–725.CrossRefGoogle Scholar
Leng, Y., Tian, X., Trask, N. and Foster, J. (2019), Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion. arXiv:1907.12031 Google Scholar
Leng, Y., Tian, X., Trask, N. and Foster, J. (2020), Asymptotically compatible reproducing kernel collocation and meshfree integration for peridynamic Navier equation. arXiv:2001.00649 Google Scholar
Lézoray, O., Ta, V.-T. and Elmoataz, A. (2010), ‘Partial differences as tools for filtering data on graphs’, Pattern Recog. Lett. 31, 2201–2213.CrossRefGoogle Scholar
Li, C. and Chen, A. (2018), ‘Numerical methods for fractional partial differential equations’, Int. J. Comput. Math. 95, 1048–1099.CrossRefGoogle Scholar
Li, S. and Liu, W. (1996), ‘Moving least-square reproducing kernel method, II: Fourier analysis’, Comput. Methods Appl. Mech. Engrg 139, 159–193.CrossRefGoogle Scholar
Li, X., Qiao, Z. and Wang, C. (2019), Convergence analysis for a stabilized linear semi-implicit numerical scheme for the nonlocal Cahn–Hilliard equation. arXiv:1902.04967 Google Scholar
Lipton, R. (2014), ‘Dynamic brittle fracture as a small horizon limit of peridynamics’, J. Elasticity 117, 21–50.CrossRefGoogle Scholar
Lipton, R. (2017), ‘Cohesive dynamics and brittle fracture’, J. Elasticity 124, 143–191.10.1007/s10659-015-9564-zCrossRefGoogle Scholar
Lipton, R., Lehoucq, R. and Sha, K. (2018), ‘Complex fracture nucleation and evolution with nonlocal elastodynamics’, J. Peridynamics Nonlocal Model. 1, 122–130.CrossRefGoogle Scholar
Liu, W., Jun, S. and Zhang, Y. (1995), ‘Reproducing kernel particle methods’, Int. J. Numer. Methods Fluids 20, 1081–1106.10.1002/fld.1650200824CrossRefGoogle Scholar
Lou, Y., Zhang, X., Osher, S. and Bertozzi, A. (2010), ‘Image recovery via nonlocal operators’, J. Sci. Comput. 42, 185–197.CrossRefGoogle Scholar
Mainardi, F. (1997), Fractional calculus. In Fractals and Fractional Calculus in Continuum Mechanics, Vol. 378 of International Centre for Mechanical Sciences (Courses and Lectures), Springer, pp. 291–348.CrossRefGoogle Scholar
Mao, Z. and Karniadakis, G. (2018), ‘A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative’, SIAM J. Numer. Anal. 56, 24–49.CrossRefGoogle Scholar
Mao, Z. and Shen, J. (2017), ‘Hermite spectral methods for fractional PDEs in unbounded domains’, SIAM J. Sci. Comput. 39, A1928–A1950.CrossRefGoogle Scholar
Mao, Z. and Shen, J. (2018), ‘Spectral element method with geometric mesh for two-sided fractional differential equations’, Adv. Comput. Math. 44, 745–771.10.1007/s10444-017-9561-9CrossRefGoogle Scholar
Mao, Z., Chen, S. and Shen, J. (2016), ‘Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations’, Appl. Numer. Math. 106, 165–181.CrossRefGoogle Scholar
McLean, W. (2000), Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press.Google Scholar
Meerschaert, M. and Sikorskii, A. (2012), Stochastic Models for Fractional Calculus, De Gruyter.Google Scholar
Meerschaert, M. and Tadjeran, C. (2004), ‘Finite difference approximations for fractional advection–dispersion flow equations’, J. Comput. Appl. Math. 172, 65–77.CrossRefGoogle Scholar
Meerschaert, M. and Tadjeran, C. (2006), ‘Finite difference approximations for two-sided space-fractional partial differential equations’, Appl. Numer. Math. 56, 80–90.CrossRefGoogle Scholar
Meerschaert, M., Benson, D. and Bäumer, B. (1999), ‘Multidimensional advection and fractional dispersion’, Phys. Rev. E 59, 5026–5028.CrossRefGoogle ScholarPubMed
Meidner, D., Pfefferer, J., Schürholz, K. and Vexler, B. (2018), ‘ $\mathrm{hp}$ -finite elements for fractional diffusion’, SIAM J. Numer. Anal. 56, 2345–2374.CrossRefGoogle Scholar
Mengesha, T. and Du, Q. (2013), ‘Analysis of a scalar nonlocal peridynamic model with sign changing kernel’, Discrete Contin. Dyn. Syst. B 18, 1415–1437.CrossRefGoogle Scholar
Mengesha, T. and Du, Q. (2014), ‘Nonlocal constrained value problems for a linear peridynamic Navier equation’, J. Elasticity 116, 27–51.CrossRefGoogle Scholar
Mengesha, T. and Du, Q. (2015), ‘On the variational limit of a class of nonlocal functionals related to peridynamics’, Nonlinearity 28, 3999–4035.CrossRefGoogle Scholar
Mengesha, T. and Du, Q. (2016), ‘Characterization of function spaces of vector fields and an application in nonlinear peridynamics’, Nonlin. Anal. Theory Methods Appl. 140, 82–111.CrossRefGoogle Scholar
Metzler, R. and Klafter, J. (2000), ‘The random walk’s guide to anomalous diffusion: A fractional dynamics approach’, Phys. Rep. 339, 77.CrossRefGoogle Scholar
Minden, V. and Ying, L. (2018), A simple solver for the fractional Laplacian in multiple dimensions. arXiv:1802.03770 Google Scholar
Molčanov, S. and Ostrovskiĭ, E. (1969), ‘Symmetric stable processes as traces of degenerate diffusion processes’, Teor. Verojatnost. i Primenen. 14, 127–130.Google Scholar
Muller, M. (1956), ‘Some continuous Monte Carlo methods for the Dirichlet problem’, Ann. Math. Statist. 27, 569–589.CrossRefGoogle Scholar
Necas, J. (1967), Les Méthodes Directes en Théorie des Équations Elliptiques, Masson.Google Scholar
Necas, J. (2016), ‘Sur une méthode pour résoudre les équations aux dérivées partielles de type elliptique, voisine de la variationnelle’, Ann. Scuola Norm. Sup. Pisa 50, 1425–1455.Google Scholar
Nochetto, R., Otárola, E. and Salgado, A. (2015), ‘A PDE approach to fractional diffusion in general domains: a priori error analysis’, Found. Comput. Math. 15, 733–791.CrossRefGoogle Scholar
Nochetto, R., von Petersdorff, T. and Zhang, C.-S. (2010), ‘A posteriori error analysis for a class of integral equations and variational inequalities’, Numer. Math. 116, 519–552.Google Scholar
Ortigueira, M. (2006), ‘Riesz potential operators and inverses via fractional centred derivatives’, Int. J. Math. Math. Sci. 2006, 48391.CrossRefGoogle Scholar
Otárola, E. and Salgado, A. (2018), ‘Sparse optimal control for fractional diffusion’, Comput. Methods Appl. Math. 18, 95–110.CrossRefGoogle Scholar
Pan, J., Ng, M. and Wang, H. (2016), ‘Fast iterative solvers for linear systems arising from time-dependent space-fractional diffusion equations’, SIAM J. Sci. Comput. 38, A2806–A2826.CrossRefGoogle Scholar
Pang, G., Chen, W. and Fu, Z. (2015), ‘Space-fractional advection–dispersion equations by the Kansa method’, J. Comput. Phys. 293, 280–296.CrossRefGoogle Scholar
Pang, G., D’Elia, M., Parks, M. and Karniadakis, G. (2019 a), nPINNs: Nonlocal physics-informed neural networks. Manuscript.Google Scholar
Pang, G., Lu, L. and Karniadakis, G. (2019 b), ‘fPINNs: Fractional physics-informed neural networks’, SIAM J. Sci. Comput. 41, A2603–A2626.CrossRefGoogle Scholar
Podlubny, I. (1999), Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, Vol. 198 of Mathematics in Science and Engineering, Academic Press.Google Scholar
Richardson, L. (1926), ‘Atmospheric diffusion shown on a distance-neighbour graph’, Proc. Roy. Soc. London A 110, 023126.Google Scholar
Ros-Oton, X. (2016), ‘Nonlocal elliptic equations in bounded domains: A survey’, Publ. Mat. 60, 3–26.Google Scholar
Ros-Oton, X. and Serra, J. (2014), ‘The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary’, J. Math. Pures Appl. 101, 275–302.CrossRefGoogle Scholar
Rosenfeld, J., Rosenfeld, S. and Dixon, W. E. (2019), ‘A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions’, J. Comput. Phys. 390, 306–322.CrossRefGoogle Scholar
Rudin, L., Osher, S. and Fatemi, E. (1992), ‘Nonlinear total variation based noise removal algorithms’, Phys. D Nonlin. Phenomena 60, 259–268.CrossRefGoogle Scholar
Sabelfeld, K. and Simonov, N. (1994), Random Walks on Boundary for Solving PDEs, VSP.CrossRefGoogle Scholar
Salazar, J. and Collins, L. (2009), ‘Two-particle dispersion in isotropic turbulent flows’, Ann. Rev. Fluid Mech. 41, 405–432.CrossRefGoogle Scholar
Samiee, M., Zayernouri, M. and Meerschaert, M. (2019 a), ‘A unified spectral method for FPDEs with two-sided derivatives, I: A fast solver’, J. Comput. Phys. 385, 225–243.CrossRefGoogle Scholar
Samiee, M., Zayernouri, M. and Meerschaert, M. (2019 b), ‘A unified spectral method for FPDEs with two-sided derivatives, II: Stability, and error analysis’, J. Comput. Phys. 385, 244–261.CrossRefGoogle Scholar
Sauter, S. and Schwab, C. (2010), Boundary element methods. In Boundary Element Methods , Vol. 39 of Springer Series in Computational Mathematics, Springer, pp. 183–287.CrossRefGoogle Scholar
Seleson, P., Du, Q. and Parks, M. (2016), ‘On the consistency between nearest-neighbor peridynamic discretizations and discretized classical elasticity models’, Comput. Methods Appl. Mech. Engrg 311, 698–722.Google Scholar
Seleson, P., Parks, M., Gunzburger, M. and Lehoucq, R. (2009), ‘Peridynamics as an upscaling of molecular dynamics’, Multiscale Model. Simul. 8, 204–227.CrossRefGoogle Scholar
Servadei, R. and Valdinoci, E. (2013), ‘Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators’, Rev. Mat. Iberoamer. 29, 1091–1126.CrossRefGoogle Scholar
Servadei, R. and Valdinoci, E. (2014), ‘On the spectrum of two different fractional operators’, Proc. Roy. Soc. Edinburgh Sect. A 144, 831–855.CrossRefGoogle Scholar
Shardlow, T. (2019), ‘A walk outside spheres for the fractional Laplacian: Fields and first eigenvalue’, Math. Comp. 88, 2767–2792.CrossRefGoogle Scholar
Shen, J. and Wang, L.-L. (2009), ‘Some recent advances on spectral methods for unbounded domains’, Commun. Comput. Phys. 5, 195–241.Google Scholar
Shen, J., Tang, T. and Wang, L.-L. (2011), Spectral Methods, Vol. 41 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Sheng, C., Shen, J., Tang, T., Wang, L.-L. and Yuan, H. (2019), Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains. arXiv:1908.10029 Google Scholar
Silling, S. (2000), ‘Reformulation of elasticity theory for discontinuities and long-range forces’, J. Mech. Phys. Solids 48, 175–209.CrossRefGoogle Scholar
Silling, S. (2010), ‘Linearized theory of peridynamic states’, J. Elasticity 99, 85–111.CrossRefGoogle Scholar
Silling, S., Epton, M., Weckner, O., Xu, J. and Askari, E. (2007), ‘Peridynamic states and constitutive modeling’, J. Elasticity 88, 151–184.CrossRefGoogle Scholar
Silvestre, L. (2017), ‘Regularity of the obstacle problem for a fractional power of the Laplace operator’, Comm. Pure Appl. Math. 60, 67–112.CrossRefGoogle Scholar
Slevinsky, R. (2019), ‘Fast and backward stable transforms between spherical harmonic expansions and bivariate Fourier series’, Appl. Comput. Harmon. Anal. 47, 585–606.CrossRefGoogle Scholar
Slevinsky, R., Montanelli, H. and Du, Q. (2018), ‘A spectral method for nonlocal diffusion operators on the sphere’, J. Comput. Phys. 372, 893–911.CrossRefGoogle Scholar
Sokolov, I., Klafter, J. and Blumen, A. (2002), ‘Fractional kinetics’, Physics Today 55, 48–54.CrossRefGoogle Scholar
Song, F. and Karniadakis, G. (2018), A universal fractional model of wall-turbulence. arXiv:1808.10276 Google Scholar
Stinga, P. (2019), User’s guide to the fractional Laplacian and the method of semigroups. In Handbook of Fractional Calculus with Applications, Vol. 2: Fractional Differential Equations (A. Kochubei and Y. Luchko, eds), De Gruyter, pp. 235–265.Google Scholar
Stinga, P. and Torrea, J. (2010), ‘Extension problem and Harnack’s inequality for some fractional operators’, Comm. Partial Differential Equations 35, 2092–2122.CrossRefGoogle Scholar
Sun, H.-G., Liu, X., Zhang, Y., Pang, G. and Garrard, R. (2017), ‘A fast semi-discrete Kansa method to solve the two-dimensional spatiotemporal fractional diffusion equation’, J. Comput. Phys. 345, 74–90.CrossRefGoogle Scholar
Swaford, B. (2001), ‘Turbulent relative dispersion’, Ann. Rev. Fluid Mech. 33, 289–317.CrossRefGoogle Scholar
Tadjeran, C., Meerschaert, M. and Scheffler, H.-P. (2006), ‘A second-order accurate numerical approximation for the fractional diffusion equation’, J. Comput. Phys. 213, 205–213.CrossRefGoogle Scholar
Tadmor, E. (2007), Filters, mollifiers and the computation of the Gibbs phenomenon. In Acta Numerica , Vol. 16, Cambridge University Press, pp. 305–378.Google Scholar
Tang, T., Wang, L.-L., Yuan, H. and Zhou, T. (2019), Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains. arXiv:1905.02476 Google Scholar
Tang, T., Yuan, H. and Zhou, T. (2018), ‘Hermite spectral collocation methods for fractional PDEs in unbounded domains’, Commun. Comput. Phys. 24, 1143–1168.CrossRefGoogle Scholar
Tao, Y., Tian, X. and Du, Q. (2017), ‘Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations’, Appl. Math. Comput. 305, 282–298.Google Scholar
Tao, Y., Tian, X. and Du, Q. (2019), ‘Nonlocal models with heterogeneous localization and their application to seamless local–nonlocal coupling’, Multiscale Model. Simul. 17, 1052–1075.CrossRefGoogle Scholar
Taylor, M. (1996), Partial Differential Equations I, Springer.CrossRefGoogle Scholar
Tian, H., Ju, L. and Du, Q. (2015), ‘Nonlocal convection–diffusion problems and finite element approximations’, Comput. Methods Appl. Mech. Engrg 289, 60–78.CrossRefGoogle Scholar
Tian, H., Ju, L. and Du, Q. (2017), ‘A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization’, Comput. Methods Appl. Mech. Engrg 320, 46–67.CrossRefGoogle Scholar
Tian, X. (2017), Nonlocal models with a finite range of nonlocal interactions. PhD thesis, Columbia University.Google Scholar
Tian, X. and Du, Q. (2013), ‘Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations’, SIAM J. Numer. Anal. 51, 3458–3482.CrossRefGoogle Scholar
Tian, X. and Du, Q. (2014), ‘Asymptotically compatible schemes and applications to robust discretization of nonlocal models’, SIAM J. Numer. Anal. 52, 1641–1665.CrossRefGoogle Scholar
Tian, X. and Du, Q. (2015), ‘Nonconforming discontinuous Galerkin methods for nonlocal variational problems’, SIAM J. Numer. Anal. 53, 762–781.CrossRefGoogle Scholar
Tian, X. and Du, Q. (2017), ‘Trace theorems for some nonlocal function spaces with heterogeneous localization’, SIAM J. Math. Anal. 49, 1621–1644.CrossRefGoogle Scholar
Tian, X. and Du, Q. (2020), ‘Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models’, SIAM Rev. 62, 199–227.CrossRefGoogle Scholar
Tian, X. and Engquist, B. (2019), ‘Fast algorithm for computing nonlocal operators with finite interaction distance’, Commun. Math. Sci. 17, 1653–1670.CrossRefGoogle Scholar
Tian, X., Du, Q. and Gunzburger, M. (2016), ‘Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains’, Adv. Comput. Math. 42, 1363–1380.CrossRefGoogle Scholar
Trask, N., You, H., Yu, Y. and Parks, M. (2019), ‘An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics’, Comput. Methods Appl. Mech. Engrg 343, 151–165.CrossRefGoogle Scholar
Valdinoci, E. (2009), ‘From the long jump random walk to the fractional Laplacian’, Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–44.Google Scholar
Vollmann, C. and Schulz, V. (2019), ‘Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion’, Comput. Visual. Sci. 20, 29–46.CrossRefGoogle Scholar
Wang, H. and Basu, T. S. (2012), ‘A fast finite difference method for two-dimensional space-fractional diffusion equations’, SIAM J. Sci. Comput. 34, A2444–A2458.CrossRefGoogle Scholar
Wang, H. and Du, N. (2013), ‘A fast finite difference method for three-dimensional time-dependent space-fractional diffusion equations and its efficient implementation’, J. Comput. Phys. 253, 50–63.CrossRefGoogle Scholar
Wang, H. and Du, N. (2014), ‘Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations’, J. Comput. Phys. 258, 305–318.CrossRefGoogle Scholar
Wang, H. and Tian, H. (2012), ‘A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model’, J. Comput. Phys. 231, 7730–7738.CrossRefGoogle Scholar
Wendland, H. (1995), ‘Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree’, Adv. Comput. Math. 4, 389–396.Google Scholar
Wendland, H. (2005), Scattered Data Approximation , Vol. 17 of Cambridge Monographs on Applied and Computational Mathematics , Cambridge University Press.Google Scholar
Witman, D., Gunzburger, M. and Peterson, J. (2017), ‘Reduced-order modeling for nonlocal diffusion problems’, Int. J. Numer. Meth. Fluids 83, 307–327.CrossRefGoogle Scholar
Xu, F., Gunzburger, M. and Burkardt, J. (2016 a), ‘A multiscale method for nonlocal mechanics and diffusion and for the approximation of discontinuous functions’, Comput. Methods Appl. Mech. Engrg 307, 117–143.CrossRefGoogle Scholar
Xu, F., Gunzburger, M., Burkardt, J. and Du, Q. (2016 b), ‘A multiscale implementation based on adaptive mesh refinement for the nonlocal peridynamics model in one dimension’, Multiscale Model. Simul. 14, 398–429.CrossRefGoogle Scholar
Xu, K. and Darve, E. (2018 a), Efficient numerical method for models driven by Lévy process via hierarchical matrices. arXiv:1812.08324 Google Scholar
Xu, K. and Darve, E. (2018 b), Spectral method for the fractional Laplacian in 2D and 3D. arXiv:1812.08325 Google Scholar
Yan, C., Cai, W. and Zeng, X. (2013), ‘A parallel method for solving Laplace equations with Dirichlet data using local boundary integral equations and random walks’, SIAM J. Sci. Comput. 35, B868–B889.CrossRefGoogle Scholar
You, H., Yu, Y. and Kamensky, D. (2019), An asymptotically compatible formulation for local-to-nonlocal coupling problems without overlapping regions. arXiv:1912.06270 Google Scholar
Zayernouri, M. and Karniadakis, G. (2014), ‘Exponentially accurate spectral and spectral element methods for fractional ODEs’, J. Comput. Phys. 257, 460–480.CrossRefGoogle Scholar
Zayernouri, M., Ainsworth, M. and Karniadakis, G. (2015 a), ‘Tempered fractional Sturm–Liouville eigenproblems’, SIAM J. Sci. Comput. 37, A1777–A1800.CrossRefGoogle Scholar
Zayernouri, M., Ainsworth, M. and Karniadakis, G. (2015 b), ‘A unified Petrov–Galerkin spectral method for fractional PDEs’, Comput. Methods Appl. Mech. Engrg 283, 1545–1569.CrossRefGoogle Scholar
Zhang, L., Sun, H.-W. and Pang, H.-K. (2015), ‘Fast numerical solution for fractional diffusion equations by exponential quadrature rule’, J. Comput. Phys. 299, 130–143.CrossRefGoogle Scholar
Zhang, W., Yang, J., Zhang, J. and Du, Q. (2017), ‘Artificial boundary conditions for nonlocal heat equations on unbounded domain’, Commun. Comput. Phys. 21, 16–39.CrossRefGoogle Scholar
Zhang, X., Gunzburger, M. and Ju, L. (2016 a), ‘Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems’, Comput. Methods Appl. Mech. Engrg 299, 401–420.CrossRefGoogle Scholar
Zhang, X., Gunzburger, M. and Ju, L. (2016 b), ‘Quadrature rules for finite element approximations of 1D nonlocal problems’, J. Comput. Phys. 310, 213–236.CrossRefGoogle Scholar
Zhang, Z. (2019), ‘Error estimates of spectral Galerkin methods for a linear fractional reaction–diffusion equation’, J. Sci. Comput. 78, 1087–1110.CrossRefGoogle Scholar
Zhou, D. and Schölkopf, B. (2005), Regularization on discrete spaces. In Proceedings of the 27th DAGM Symposium , Vol. 3663 of Lecture Notes in Computer Science, Springer, pp. 361–36.Google Scholar
Zhou, K. and Du, Q. (2010), ‘Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions’, SIAM J. Numer. Anal. 48, 1759–1780.CrossRefGoogle Scholar