Skip to main content
SpringerLink
Log in

Numerical methods for fractional diffusion

  • Special Issue FEM Symposium 2017
  • Published:
Computing and Visualization in Science

Abstract

We present three schemes for the numerical approximation of fractional diffusion, which build on different definitions of such a non-local process. The first method is a PDE approach that applies to the spectral definition and exploits the extension to one higher dimension. The second method is the integral formulation and deals with singular non-integrable kernels. The third method is a discretization of the Dunford–Taylor formula. We discuss pros and cons of each method, error estimates, and document their performance with a few numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Abe, S., Thurner, S.: Anomalous diffusion in view of Einstein’s 1905 theory of Brownian motion. Phys. A Stat. Mech. Appl. 356(2–4), 403–407 (2005)

    Article  Google Scholar 

  2. Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)

  3. Acosta, G., Bersetche, F., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. arXiv:1705.09815v1 (2017)

  4. 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(4), 784–816 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  5. Acosta, G., Borthagaray, J.P.: A fractional Laplace equation: regularity of solutions and finite element approximations. SIAM J. Numer. Anal. 55(2), 472–495 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Acosta, G., Borthagaray, J.P., Bruno, O., Maas, M.: Regularity theory and high order numerical methods for one-dimensional fractional-Laplacian equations. Math. Comput. (2017). https://doi.org/10.1090/mcom/3276

  7. Acosta, G., Borthagaray, J.P., Heuer, N.: Finite element approximations for the nonhomogeneous fractional Dirichlet problem. arXiv:1709.06592v1 (2017)

  8. Ainsworth, M., Glusa, C.: Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains. arXiv:1708.01923v1 (2017)

  9. Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Antil, H., Otárola, E.: An a posteriori error analysis for an optimal control problem involving the fractional Laplacian. IMA J. Numer. Anal. 38(1), 198–266 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Antil, H., Otárola, E., Salgado, A.J.: Optimization with respect to order in a fractional diffusion model: analysis, approximation and algorithm aspects. arXiv:1612.08982v1 (2016)

  12. Antil, H., Otárola, E., Salgado, A.J.: A space-time fractional optimal control problem: analysis and discretization. SIAM J. Control Optim. 54(3), 1295–1328 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Eng. 61(1), 1–40 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  14. Babuška, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.-L. (eds.) Handbook of Numerical Analysis, vol. II, pp. 641–787. North-Holland, Amsterdam (1991)

    Google Scholar 

  15. Bacuta, C., Bramble, J.H., Pasciak, J.E.: New interpolation results and applications to finite element methods for elliptic boundary value problems. East West J. Numer. Math. 3, 179–198 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Bangerth, W., Hartmann, R., Kanschat, G.: deal. II—Diferential Equations Analysis Library. Technical Reference, http://www.dealii.org

  17. Bangerth, W., Hartmann, R., Kanschat, G.: deal. II—a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33(4), Art. 24, 27 (2007)

  18. Banjai, L., Melenk, J.M., Nochetto, R.H., Otárola, E., Salgado, A.J., Schwab, C.: Tensor FEM for spectral fractional diffusion. arXiv:1707.07367v1 (2017)

  19. Bertoin, J.: Lévy Processes, Volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  20. Birman, M.Š., Solomjak, M.Z.: Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve. Leningrad. University of Leningrad (1980)

  21. Bonito, A., Guermond, J.-L., Luddens, F.: Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains. J. Math. Anal. Appl. 408(2), 498–512 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bonito, A., Lei, W., Pasciak, J.E.: The approximation of parabolic equations involving fractional powers of elliptic operators. J. Comput. Appl. Math. 315, 32–48 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of space-time fractional parabolic equations. Comput. Methods Appl. Math. 17(4), 679–705 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of the integral fractional Laplacian. arXiv:1707.04290v1 (2017)

  25. Bonito, A., Pasciak, J.: Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84(295), 2083–2110 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bonito, A., Pasciak, J.E.: Numerical approximation of fractional powers of regularly accretive operators. IMA J. Numer. Anal. 37(3), 1245–1273 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Borthagaray, J.P., Ciarlet, P. Jr.: Nonlocal models for interface problems between dielectrics and metamaterials. In: 11th International Congress on Engineered Material Platforms for Novel Wave Phenomena (2017)

  28. Borthagaray, J.P., Del Pezzo, L.M., Martínez, S.: Finite element approximation for the fractional eigenvalue problem. arXiv:1603.00317v2 (2017)

  29. Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Optimal Control and Partial, Differential Equations, pp. 439–455. IOS Press, Amsterdam (2001)

  30. Brändle, C., Colorado, E., de Pablo, A., Sánchez, U.: A concave–convex elliptic problem involving the fractional Laplacian. Proc. R. Soc. Edinb. Sect. A 143(1), 39–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, Volume 15 of Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)

    Book  Google Scholar 

  32. Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439(7075), 462–465 (2006)

    Article  Google Scholar 

  33. Bucur, C., Valdinoci, E.: Nonlocal Diffusion and Applications, Volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, Unione Matematica Italiana, Berlin, Bologna (2016)

    Google Scholar 

  34. Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Caffarelli, L., Figalli, A.: Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angew. Math. 680, 191–233 (2013)

    MathSciNet  MATH  Google Scholar 

  36. Caffarelli, L., Stinga, P.: Fractional elliptic equations, Caccioppoli estimates, and regularity. Annales de l’Institut Henri Poincare (C) Non Linear Analysis 33, 767–807 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial. Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171(3), 1903–1930 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Partial Differ. Equ. 36(8), 1353–1384 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Carmichael, B., Babahosseini, H., Mahmoodi, S.N., Agah, M.: The fractional viscoelastic response of human breast tissue cells. Phys. Biol. 12(4), 046001 (2015)

    Article  Google Scholar 

  42. Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002)

    Article  Google Scholar 

  43. Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion: a posteriori error analysis. J. Comput. Phys. 293, 339–358 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: Multilevel methods for nonuniformly elliptic operators and fractional diffusion. Math. Comput. 85(302), 2583–2607 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Chen, Z.Q., Song, R.: Hardy inequality for censored stable processes. Tohoku Math. J. (2) 55(3), 439–450 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ciarlet Jr., P.: Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces. J. Numer. Math. 21(3), 173–180 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Čiegis, R., Starikovičius, V., Margenov, S., Kriauzienė, R.: Parallel solvers for fractional power diffusion problems. Concurr. Comput. Pract. Exp. 29, e4216 (2017)

    Article  Google Scholar 

  48. Costabel, M., Dauge, M.: General edge asymptotics of solutions of second-order elliptic boundary value problems I. Proc. R. Soc. Edinb. Sect. A Math. 123(1), 109–155 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  49. Cushman, J., Glinn, T.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Trans. Porous Media 13, 123–138 (1993)

    Article  Google Scholar 

  50. Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

    Book  Google Scholar 

  51. D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66(7), 1245–1260 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  53. Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  54. Duoandikoetxea, J.: Fourier Analysis, Volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001). Translated and revised from the 1995 Spanish original by David Cruz-Uribe

  55. Durán, R.G., Lombardi, A.L.: Error estimates on anisotropic \(Q_1\) elements for functions in weighted Sobolev spaces. Math. Comput. 74(252), 1679–1706 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  56. Dyda, B.: A fractional order Hardy inequality. Ill. J. Math. 48(2), 575–588 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  57. Dyda, B., Kuznetsov, A., Kwaśnicki, M.: Eigenvalues of the fractional Laplace operator in the unit ball. J. Lond. Math. Soc. 95(2), 500–518 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  58. Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Hildebrandt, S., Leis, R. (eds.) Partial Differential Equations and Calculus of Variations, pp. 142–155. Springer, Berlin (1988)

    Chapter  Google Scholar 

  59. Einstein, A.: Investigations on the theory of the Brownian movement. Dover Publications Inc., New York, Edited with notes by R. Fürth, Translated by A. D. Cowper (1956)

  60. Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Faermann, B.: Localization of the Aronszajn–Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case. Numer. Math. 92(3), 467–499 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  62. Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  63. Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361(7), 3829–3850 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  64. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag–Leffler Functions, Related Topics and Applications. Springer Monographs in Mathematics. Springer, Heidelberg (2014)

    MATH  Google Scholar 

  65. Grisvard, P.: Elliptic problems in nonsmooth domains, Volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, Reprint of the 1985 original [MR0775683], With a foreword by Susanne C. Brenner (2011)

  66. Grubb, G.: Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  67. Grubb, G.: Spectral results for mixed problems and fractional elliptic operators. J. Math. Anal. Appl. 421(2), 1616–1634 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  68. Hörmander, L.: Ch. II, Boundary problems for “classical” pseudo-differential operators. http://www.math.ku.dk/~grubb/LH65.pdf (1965)

  69. Huang, Y., Oberman, A.M.: Numerical methods for the fractional Laplacian: a finite difference-quadrature approach. SIAM J. Numer. Anal. 52(6), 3056–3084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  70. Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38(1), A146–A170 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  71. Jochmann, F.: An \({H}^s\)-regularity result for the gradient of solutions to elliptic equations with mixed boundary conditions. J. Math. Anal. Appl. 238, 429–450 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  72. Kato, T.: Note on fractional powers of linear operators. Proc. Jpn. Acad. 36, 94–96 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  73. Kato, T.: Fractional powers of dissipative operators. J. Math. Soc. Jpn. 13, 246–274 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  74. Kellogg, R.B.: Interpolation between subspaces of a Hilbert space. Technical report, University of Maryland, Institute for Fluid Dynamics and Applied Mathematics, Technical note BN-719 (1971)

  75. Kilpeläinen, T.: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. AI Math. 19(1), 95–113 (1994)

    MathSciNet  MATH  Google Scholar 

  76. Krasnosel’skiĭ, M.A., Rutickiĭ, J.B.: Convex functions and Orlicz spaces. Translated from the first Russian edition by Leo F. Boron. P. Noordhoff Ltd., Groningen (1961)

  77. Kufner, A.: Weighted Sobolev spaces. Wiley, New York (1985). Translated from the Czech

  78. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)

    MathSciNet  MATH  Google Scholar 

  79. Kyprianou, A., Osojnik, A., Shardlow, T.: Unbiased walk-on-spheres’ Monte Carlo methods for the fractional Laplacian. IMA J. Numer. Anal. (2017). https://doi.org/10.1093/imanum/drx042

  80. Landkof, N.S.: Foundations of modern potential theory. Springer, New York (1972). Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180

  81. Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4), 298–305 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  82. Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52(2), 129–145 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  83. Lunardi, A.: Interpolation theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 2nd edn. Edizioni della Normale, Pisa (2009)

  84. Lund, J., Bowers, K.L.: Sinc Methods for Quadrature and Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)

    Book  MATH  Google Scholar 

  85. McCay, B.M., Narasimhan, M.N.L.: Theory of nonlocal electromagnetic fluids. Arch. Mech. 33(3), 365–384 (1981)

    MathSciNet  MATH  Google Scholar 

  86. McIntosh, A.: The square root problem for elliptic operators: a survey. In: Fujita, H., Ikebe, T., Kuroda, S.T. (eds.) Functional-Analytic Methods for Partial Differential Equations (Tokyo, 1989), Volume 1450 of Lecture Notes in Mathematics, pp. 122–140. Springer, Berlin (1990)

    Google Scholar 

  87. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  88. Meidner, D., Pfefferer, J., Schürholz, K., Vexler, B.: \(hp\)-finite elements for fractional diffusion. arXiv:1706.04066v1 (2017)

  89. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A 37(31), R161–R208 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  90. Morin, P., Nochetto, R.H., Siebert, K.G.: Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comput. 72(243), 1067–1097 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  91. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  92. Musina, R., Nazarov, A.I.: On fractional Laplacians. Commun. Partial Differ. Equ. 39(9), 1780–1790 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  93. Nazarov, S., Plamenevsky, B.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Expositions in Mathematics. De Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  94. Nochetto, R.H., Otárola, E., Salgado, A.J.: Convergence rates for the classical, thin and fractional elliptic obstacle problems. Philos. Trans. Roy. Soc. A 373(2050), 20140449 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  95. Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  96. Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to numerical fractional diffusion. In: Proceedings of the 8th International Congress on Industrial and Applied Mathematics, pp. 211–236. Higher Ed. Press, Beijing (2015)

  97. Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to space-time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  98. Nochetto, R.H., Otárola, E., Salgado, A.J.: Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132(1), 85–130 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  99. Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R., Kunoth, A. (eds.) Multiscale Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)

    Chapter  MATH  Google Scholar 

  100. Nochetto, R.H., Veeser, A.: Primer of adaptive finite element methods. In: Naldi, G., Russo, G. (eds.) Multiscale and Adaptivity: Modeling, Numerics and Applications. CIME Lectures. Springer, Berlin (2011)

    Google Scholar 

  101. Nochetto, R.H., von Petersdorff, T., Zhang, C.-S.: A posteriori error analysis for a class of integral equations and variational inequalities. Numer. Math. 116(3), 519–552 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  102. Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters, Ltd., Wellesley (1997). Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]

  103. Otárola, E.: A PDE approach to numerical fractional diffusion. ProQuest LLC, Ann Arbor (2014). Thesis Ph.D., University of Maryland, College Park

  104. Otárola, E.: A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains. ESAIM Math. Model. Numer. Anal. 51(4), 1473–1500 (2017)

    MathSciNet  MATH  Google Scholar 

  105. Otárola, E., Salgado, A.J.: Finite element approximation of the parabolic fractional obstacle problem. SIAM J. Numer. Anal. 54(4), 2619–2639 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  106. Otárola, E., Salgado, A.J.: Regularity of solutions to space–time fractional wave equations: a PDE approach. arXiv:1711.06186 (2017)

  107. Otárola, E., Salgado, A.J.: Sparse optimal control for fractional diffusion. Comput. Math. Appl. Math. 18(1), 95–110 (2018)

    MathSciNet  MATH  Google Scholar 

  108. Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Math. 60(1), 3–26 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  109. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101(3), 275–302 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  110. Ros-Oton, X., Serra, J.: Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete Contin. Dyn. Syst. 35(5), 2131–2150 (2015)

    MathSciNet  MATH  Google Scholar 

  111. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  112. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993). Theory and Applications, Edited and with a foreword by S. M. Nikol\(\prime \)skiĭ, Translated from the 1987 Russian original, Revised by the authors

  113. Sauter, S.A., Schwab, C.: Boundary Element Methods, Volume 39 of Springer Series in Computational Mathematics. Springer, Berlin (2011). Translated and expanded from the 2004 German original

  114. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  115. Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. Sect. A 144(4), 831–855 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  116. Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  117. Sims, D., Southall, E., Humphries, N., Hays, G., Bradshaw, C., Pitchford, J., James, A., Ahmed, M., Brierley, A., Hindell, M., Morritt, D., Musyl, M., Righton, D., Shepard, E., Wearmouth, V., Wilson, R., Witt, M., Metcalfe, J.: Scaling laws of marine predator search behaviour. Nature 451(7182), 1098–1102 (2008)

    Article  Google Scholar 

  118. Sprekels, J., Valdinoci, E.: A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation. SIAM J. Control Optim. 55(1), 70–93 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  119. Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  120. Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces, Volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin (2007)

    Google Scholar 

  121. Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)

  122. Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces, Volume of Lecture Notes in Mathematics. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  123. Višik, M.I., Èskin, G.I.: Elliptic convolution equations in a bounded region and their applications. Uspehi Mat. Nauk. 22:1(133), 15–76 (1967)

    MathSciNet  Google Scholar 

  124. Yosida, K.: Functional Analysis (Die Grundlehren der mathematischen Wissenschaften, Band 123), 2nd edn. Springer, New York (1968)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Andrea Bonito

  2. IMAS - CONICET and Departamento de Matemática, FCEyN - Universidad de Buenos Aires, Buenos Aires, Argentina

    Juan Pablo Borthagaray

  3. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  4. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaiso, Chile

    Enrique Otárola

  5. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Abner J. Salgado

Authors
  1. Andrea Bonito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juan Pablo Borthagaray
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ricardo H. Nochetto
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Enrique Otárola
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Abner J. Salgado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ricardo H. Nochetto.

Additional information

Communicated by Thomas Apel.

AB is supported in part by NSF grant DMS-1254618. JPB has been partially supported by a CONICET doctoral fellowship. RHN has been supported in part by NSF grant DMS-1411808. EO has been supported in part by CONICYT through project FONDECYT 3160201. AJS is supported by NSF grant DMS-1418784.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonito, A., Borthagaray, J.P., Nochetto, R.H. et al. Numerical methods for fractional diffusion. Comput. Visual Sci. 19, 19–46 (2018). https://doi.org/10.1007/s00791-018-0289-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-018-0289-y

Search

Navigation