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Computing and Visualization in Science

, Volume 19, Issue 5–6, pp 19–46 | Cite as

Numerical methods for fractional diffusion

  • Andrea Bonito
  • Juan Pablo Borthagaray
  • Ricardo H. NochettoEmail author
  • Enrique Otárola
  • Abner J. Salgado
Special Issue FEM Symposium 2017
First Online:

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.

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References

  1. 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)CrossRefGoogle Scholar
  2. 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)Google Scholar
  3. 3.
    Acosta, G., Bersetche, F., Borthagaray, J.P.: Finite element approximations for fractional evolution problems. arXiv:1705.09815v1 (2017)
  4. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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. 7.
    Acosta, G., Borthagaray, J.P., Heuer, N.: Finite element approximations for the nonhomogeneous fractional Dirichlet problem. arXiv:1709.06592v1 (2017)
  8. 8.
    Ainsworth, M., Glusa, C.: Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains. arXiv:1708.01923v1 (2017)
  9. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 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. 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)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bangerth, W., Hartmann, R., Kanschat, G.: deal. II—Diferential Equations Analysis Library. Technical Reference, http://www.dealii.org
  17. 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)Google Scholar
  18. 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. 19.
    Bertoin, J.: Lévy Processes, Volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1996)Google Scholar
  20. 20.
    Birman, M.Š., Solomjak, M.Z.: Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve. Leningrad. University of Leningrad (1980)Google Scholar
  21. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bonito, A., Lei, W., Pasciak, J.E.: Numerical approximation of the integral fractional Laplacian. arXiv:1707.04290v1 (2017)
  25. 25.
    Bonito, A., Pasciak, J.: Numerical approximation of fractional powers of elliptic operators. Math. Comput. 84(295), 2083–2110 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Bonito, A., Pasciak, J.E.: Numerical approximation of fractional powers of regularly accretive operators. IMA J. Numer. Anal. 37(3), 1245–1273 (2017)MathSciNetzbMATHGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Borthagaray, J.P., Del Pezzo, L.M., Martínez, S.: Finite element approximation for the fractional eigenvalue problem. arXiv:1603.00317v2 (2017)
  29. 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)Google Scholar
  30. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. 32.
    Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439(7075), 462–465 (2006)CrossRefGoogle Scholar
  33. 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. 34.
    Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Caffarelli, L., Figalli, A.: Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angew. Math. 680, 191–233 (2013)MathSciNetzbMATHGoogle Scholar
  36. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial. Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171(3), 1903–1930 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 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)CrossRefGoogle Scholar
  43. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Chen, Z.Q., Song, R.: Hardy inequality for censored stable processes. Tohoku Math. J. (2) 55(3), 439–450 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Ciarlet Jr., P.: Analysis of the Scott–Zhang interpolation in the fractional order Sobolev spaces. J. Numer. Math. 21(3), 173–180 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Čiegis, R., Starikovičius, V., Margenov, S., Kriauzienė, R.: Parallel solvers for fractional power diffusion problems. Concurr. Comput. Pract. Exp. 29, e4216 (2017)CrossRefGoogle Scholar
  48. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Cushman, J., Glinn, T.: Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Trans. Porous Media 13, 123–138 (1993)CrossRefGoogle Scholar
  50. 50.
    Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)CrossRefGoogle Scholar
  51. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 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-UribeGoogle Scholar
  55. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Dyda, B.: A fractional order Hardy inequality. Ill. J. Math. 48(2), 575–588 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  57. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 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)CrossRefGoogle Scholar
  59. 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)Google Scholar
  60. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Getoor, R.K.: First passage times for symmetric stable processes in space. Trans. Am. Math. Soc. 101, 75–90 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361(7), 3829–3850 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 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)zbMATHGoogle Scholar
  65. 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)Google Scholar
  66. 66.
    Grubb, G.: Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Grubb, G.: Spectral results for mixed problems and fractional elliptic operators. J. Math. Anal. Appl. 421(2), 1616–1634 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Hörmander, L.: Ch. II, Boundary problems for “classical” pseudo-differential operators. http://www.math.ku.dk/~grubb/LH65.pdf (1965)
  69. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Kato, T.: Note on fractional powers of linear operators. Proc. Jpn. Acad. 36, 94–96 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Kato, T.: Fractional powers of dissipative operators. J. Math. Soc. Jpn. 13, 246–274 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  74. 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)Google Scholar
  75. 75.
    Kilpeläinen, T.: Weighted Sobolev spaces and capacity. Ann. Acad. Sci. Fenn. Ser. AI Math. 19(1), 95–113 (1994)MathSciNetzbMATHGoogle Scholar
  76. 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)Google Scholar
  77. 77.
    Kufner, A.: Weighted Sobolev spaces. Wiley, New York (1985). Translated from the CzechGoogle Scholar
  78. 78.
    Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)MathSciNetzbMATHGoogle Scholar
  79. 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. 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 180Google Scholar
  81. 81.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268(4), 298–305 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Lubich, C.: Convolution quadrature and discretized operational calculus I. Numer. Math. 52(2), 129–145 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 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)Google Scholar
  84. 84.
    Lund, J., Bowers, K.L.: Sinc Methods for Quadrature and Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)zbMATHCrossRefGoogle Scholar
  85. 85.
    McCay, B.M., Narasimhan, M.N.L.: Theory of nonlocal electromagnetic fluids. Arch. Mech. 33(3), 365–384 (1981)MathSciNetzbMATHGoogle Scholar
  86. 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. 87.
    McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  88. 88.
    Meidner, D., Pfefferer, J., Schürholz, K., Vexler, B.: \(hp\)-finite elements for fractional diffusion. arXiv:1706.04066v1 (2017)
  89. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  90. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Musina, R., Nazarov, A.I.: On fractional Laplacians. Commun. Partial Differ. Equ. 39(9), 1780–1790 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  93. 93.
    Nazarov, S., Plamenevsky, B.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Expositions in Mathematics. De Gruyter, Berlin (1994)zbMATHCrossRefGoogle Scholar
  94. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  95. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 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)Google Scholar
  97. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  98. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  99. 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)CrossRefzbMATHGoogle Scholar
  100. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  102. 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)]Google Scholar
  103. 103.
    Otárola, E.: A PDE approach to numerical fractional diffusion. ProQuest LLC, Ann Arbor (2014). Thesis Ph.D., University of Maryland, College ParkGoogle Scholar
  104. 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)MathSciNetzbMATHGoogle Scholar
  105. 105.
    Otárola, E., Salgado, A.J.: Finite element approximation of the parabolic fractional obstacle problem. SIAM J. Numer. Anal. 54(4), 2619–2639 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  106. 106.
    Otárola, E., Salgado, A.J.: Regularity of solutions to space–time fractional wave equations: a PDE approach. arXiv:1711.06186 (2017)
  107. 107.
    Otárola, E., Salgado, A.J.: Sparse optimal control for fractional diffusion. Comput. Math. Appl. Math. 18(1), 95–110 (2018)MathSciNetzbMATHGoogle Scholar
  108. 108.
    Ros-Oton, X.: Nonlocal elliptic equations in bounded domains: a survey. Publ. Math. 60(1), 3–26 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  109. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  110. 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)MathSciNetzbMATHGoogle Scholar
  111. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 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 authorsGoogle Scholar
  113. 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 originalGoogle Scholar
  114. 114.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. Sect. A 144(4), 831–855 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  116. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  117. 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)CrossRefGoogle Scholar
  118. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  119. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  120. 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. 121.
    Taylor, M.E.: Pseudodifferential Operators. Princeton Mathematical Series, vol. 34. Princeton University Press, Princeton (1981)Google Scholar
  122. 122.
    Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces, Volume of Lecture Notes in Mathematics. Springer, Berlin (2000)zbMATHCrossRefGoogle Scholar
  123. 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)MathSciNetGoogle Scholar
  124. 124.
    Yosida, K.: Functional Analysis (Die Grundlehren der mathematischen Wissenschaften, Band 123), 2nd edn. Springer, New York (1968)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Andrea Bonito
    • 1
  • Juan Pablo Borthagaray
    • 2
  • Ricardo H. Nochetto
    • 3
    Email author
  • Enrique Otárola
    • 4
  • Abner J. Salgado
    • 5
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.IMAS - CONICET and Departamento de MatemáticaFCEyN - Universidad de Buenos AiresBuenos AiresArgentina
  3. 3.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  4. 4.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile
  5. 5.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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