Foundations of Computational Mathematics

, Volume 15, Issue 3, pp 733–791 | Cite as

A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis

  • Ricardo H. NochettoEmail author
  • Enrique Otárola
  • Abner J. Salgado


The purpose of this work is to study solution techniques for problems involving fractional powers of symmetric coercive elliptic operators in a bounded domain with Dirichlet boundary conditions. These operators can be realized as the Dirichlet-to-Neumann map for a degenerate/singular elliptic problem posed on a semi-infinite cylinder, which we analyze in the framework of weighted Sobolev spaces. Motivated by the rapid decay of the solution to this problem, we propose a truncation that is suitable for numerical approximation. We discretize this truncation using first degree tensor product finite elements. We derive a priori error estimates in weighted Sobolev spaces. The estimates exhibit optimal regularity but suboptimal order for quasi-uniform meshes. For anisotropic meshes instead, they are quasi-optimal in both order and regularity. We present numerical experiments to illustrate the method’s performance.


Fractional diffusion Finite elements Nonlocal operators Degenerate and singular equations Second-order elliptic operators Anisotropic elements 

Mathematics Subject Classification

35S15 65R20 65N12 65N30 



This work is supported by NSF Grants DMS-1109325 and DMS-0807811. A.J.S. is also supported by NSF Grant DMS-1008058 and an AMS-Simons Grant. E.O. is supported by the Conicyt–Fulbright Fellowship Beca Igualdad de Oportunidades.


  1. 1.
    M. Abramowitz and I.A. Stegun. 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
  2. 2.
    G. Acosta. Lagrange and average interpolation over 3D anisotropic elements. J. Comput. Appl. Math., 135(1):91–109, 2001.Google Scholar
  3. 3.
    R.A. Adams. Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65.Google Scholar
  4. 4.
    T. Apel. Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN Math. Model. Numer. Anal., 33(6):1149–1185, 1999.Google Scholar
  5. 5.
    O.G. Bakunin. Turbulence and diffusion. Springer Series in Synergetics. Springer-Verlag, Berlin, 2008. Scaling versus equations.Google Scholar
  6. 6.
    W. Bangerth, R. Hartmann, and G. Kanschat. deal.II–differential equations analysis library. Technical Reference:
  7. 7.
    W. Bangerth, R. Hartmann, and G. Kanschat. deal.II–a general-purpose object-oriented finite element library. ACM Trans. Math. Software, 33(4):Art. 24, 27, 2007.Google Scholar
  8. 8.
    P.W. Bates. On some nonlocal evolution equations arising in materials science. In Nonlinear dynamics and evolution equations, volume 48 of Fields Inst. Commun., pages 13–52. Amer. Math. Soc., Providence, RI, 2006.Google Scholar
  9. 9.
    Z. Belhachmi, Ch. Bernardi, and S. Deparis. Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math., 105(2):217–247, 2006.Google Scholar
  10. 10.
    D.A. Benson, S.W. Wheatcraft, and M.M. Meerschaert. Application of a fractional advection-dispersion equation. Water Resources Res., 36:91–109, 2000.Google Scholar
  11. 11.
    C. Bernardi. Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal., 26(5):1212–1240, 1989.Google Scholar
  12. 12.
    J. Bertoin. Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.Google Scholar
  13. 13.
    M.Š. Birman and M.Z. Solomjak. Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve. Leningrad. Univ., Leningrad, 1980.Google Scholar
  14. 14.
    K. Bogdan, K. Burdy, and Chen Z.Q. Censored stable processes. Probab. Theory Related Fields, 127(R-2):89–152, 2003.Google Scholar
  15. 15.
    A. Bonito and J.E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Math. Comp. (to appear), 2014.Google Scholar
  16. 16.
    C. Brändle, E. Colorado, A. de Pablo, and U. Sánchez. A concave-convex elliptic problem involving the fractional Laplacian. Proc. Roy. Soc. Edinburgh Sect. A, 143(1):39–71, 2013.Google Scholar
  17. 17.
    S.C. Brenner and L.R. Scott. The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York, third edition, 2008.Google Scholar
  18. 18.
    Kevin Burrage, Nicholas Hale, and David Kay. An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput., 34(4):A2145–A2172, 2012.Google Scholar
  19. 19.
    X. Cabré and Y. Sire. Nonlinear equations for fractional Laplacians ii: Existence, uniqueness and qualitative properties of solutions. arXiv:1111.0796v1, 2011.
  20. 20.
    X. Cabré and J. Tan. Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math., 224(5):2052–2093, 2010.Google Scholar
  21. 21.
    L. Caffarelli and L. Silvestre. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32(7–9):1245–1260, 2007.Google Scholar
  22. 22.
    A. Capella, J. Dávila, L. Dupaigne, and Y. Sire. Regularity of radial extremal solutions for some non-local semilinear equations. Comm. Partial Differential Equations, 36(8):1353–1384, 2011.Google Scholar
  23. 23.
    P. Carr, H. Geman, D.B. Madan, and M. Yor. The fine structure of asset returns: An empirical investigation. Journal of Business, 75:305–332, 2002.Google Scholar
  24. 24.
    P.G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)].Google Scholar
  25. 25.
    P.G. Ciarlet and P.-A. Raviart. Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg., 1:217–249, 1972.Google Scholar
  26. 26.
    P. Clément. Approximation by finite element functions using local regularization. RAIRO Analyse Numérique, 9(2):77–84, 1975.Google Scholar
  27. 27.
    J. Cushman and T. Glinn. Nonlocal dispersion in media with continuously evolving scales of heterogeneity. Trans. Porous Media, 13:123–138, 1993.Google Scholar
  28. 28.
    E. Di Nezza, G. Palatucci, and E. Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math., 136(5):521–573, 2012.Google Scholar
  29. 29.
    T. Dupont and R. Scott. Polynomial approximation of functions in sobolev spaces. Math. Comp., 34:441–463, 1980.Google Scholar
  30. 30.
    R.G. Durán and A.L. Lombardi. Error estimates on anisotropic \(Q_1\) elements for functions in weighted Sobolev spaces. Math. Comp., 74(252):1679–1706 (electronic), 2005.Google Scholar
  31. 31.
    R.G. Durán, A.L. Lombardi, and M.I. Prieto. Superconvergence for finite element approximation of a convection-diffusion equation using graded meshes. IMA Journal of Numerical Analysis, 32(2):511–533, 2012.Google Scholar
  32. 32.
    R.G. Durán and F. López García. Solutions of the divergence and Korn inequalities on domains with an external cusp. Ann. Acad. Sci. Fenn. Math., 35(2):421–438, 2010.Google Scholar
  33. 33.
    G. Duvaut and J.-L. Lions. Inequalities in mechanics and physics. Springer-Verlag, Berlin, 1976. Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219.Google Scholar
  34. 34.
    A.C. Eringen. Nonlocal continuum field theories. Springer-Verlag, New York, 2002.Google Scholar
  35. 35.
    L.C. Evans. Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.Google Scholar
  36. 36.
    E. B. Fabes, C.E. Kenig, and R.P. Serapioni. The local regularity of solutions of degenerate elliptic equations. Comm. Partial Differential Equations, 7(1):77–116, 1982.Google Scholar
  37. 37.
    D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.Google Scholar
  38. 38.
    G. Gilboa and S. Osher. Nonlocal operators with applications to image processing. Multiscale Model. Simul., 7(3):1005–1028, 2008.Google Scholar
  39. 39.
    V. Gol’dshtein and A. Ukhlov. Weighted Sobolev spaces and embedding theorems. Trans. Amer. Math. Soc., 361(7):3829–3850, 2009.Google Scholar
  40. 40.
    Q.Y. Guan and Z.M. Ma. Reflected symmetric \(\alpha \)-stable processes and regional fractional laplacian. Probab. Theory Related Fields, 134(2):649–694, 2006.Google Scholar
  41. 41.
    J. Heinonen, T. Kilpeläinen, and O. Martio. Nonlinear potential theory of degenerate elliptic equations. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1993. Oxford Science Publications.Google Scholar
  42. 42.
    M. Ilic, F. Liu, I. Turner, and V. Anh. Numerical approximation of a fractional-in-space diffusion equation. I. Fract. Calc. Appl. Anal., 8(3):323–341, 2005.Google Scholar
  43. 43.
    M. Ilic, F. Liu, I. Turner, and V. Anh. Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal., 9(4):333–349, 2006.Google Scholar
  44. 44.
    V.G. Korneev. The construction of variational difference schemes of a high order of accuracy. Vestnik Leningrad. Univ., 25(19):28–40, 1970. (In Russian).Google Scholar
  45. 45.
    V.G. Korneev and S.E. Ponomarev. Application of curvilinear finite elements in schemes for solution of \(2n\)-order linear elliptic equations. I. Čisl. Metody Meh. Splošn. Sredy, 5(5):78–97, 1974. (In Russian).Google Scholar
  46. 46.
    A. Kufner. Weighted Sobolev spaces. A Wiley-Interscience Publication. John Wiley & Sons Inc., New York, 1985. Translated from the Czech.Google Scholar
  47. 47.
    N.S. Landkof. Foundations of modern potential theory. Springer-Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.Google Scholar
  48. 48.
    J.-L. Lions and E. Magenes. Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York, 1972. Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181.Google Scholar
  49. 49.
    B.M. McCay and M.N.L. Narasimhan. Theory of nonlocal electromagnetic fluids. Arch. Mech. (Arch. Mech. Stos.) 33(3):365–384, 1981.Google Scholar
  50. 50.
    W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge, 2000.Google Scholar
  51. 51.
    K.S. Miller and S.G. Samko. Completely monotonic functions. Integral Transform. Spec. Funct., 12(4):389–402, 2001.Google Scholar
  52. 52.
    B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972.Google Scholar
  53. 53.
    R.H. Nochetto, E. Otárola, and A.J Salgado. A pde approach to space-time fractional diffusion. arXiv:1404.0068, 2014.
  54. 54.
    R.H. Nochetto, E. Otárola, and A.J. Salgado. Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and appications. arXiv:1402.1916, 2014.
  55. 55.
    G. Savaré. Regularity and perturbation results for mixed second order elliptic problems. Comm. Partial Differential Equations, 22(5–6):869–899, 1997.Google Scholar
  56. 56.
    L.R. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54(190):483–493, 1990.Google Scholar
  57. 57.
    R. Servadei and E. Valdinoci. On the spectrum of two different fractional operators. preprint, 2012.Google Scholar
  58. 58.
    S.A. Silling. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids, 48(1):175–209, 2000.Google Scholar
  59. 59.
    E.M. Stein. Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.Google Scholar
  60. 60.
    P.R. Stinga and J.L. Torrea. Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differential Equations, 35(11):2092–2122, 2010.Google Scholar
  61. 61.
    L. Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin, 2007.Google Scholar
  62. 62.
    Q. Yang, I. Turner, F. Liu, and M. Ilić. Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. SIAM J. Sci. Comput., 33(3):1159–1180, 2011.Google Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  • Ricardo H. Nochetto
    • 1
    Email author
  • Enrique Otárola
    • 2
  • Abner J. Salgado
    • 2
    • 3
  1. 1.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsUniversity of TennesseKnoxvilleUSA

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