A PDE approach to fractional diffusion: a space-fractional wave equation

Abstract

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order \(s \in (0,1)\), of symmetric, coercive, linear, elliptic, second-order operators in bounded domains \(\varOmega \). We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder \(\mathcal {C}= \varOmega \times (0,\infty )\). We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in \(\varOmega \) with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in \(\varOmega \) with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains \(\varOmega \subset {\mathbb {R}}^2\). We discuss implementation details and report several numerical examples.

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

Fig. 1
Fig. 2

References

  1. 1.

    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Dover Publications, Inc., New York (1992)

  2. 2.

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

  3. 3.

    Athanasopoulos, I., Caffarelli, L.A.: Continuity of the temperature in boundary heat control problems. Adv. Math. 224(1), 293–315 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Băcuţă, C., Li, H., Nistor, V.: Differential operators on domains with conical points: precise uniform regularity estimates. Rev. Roum. Math. Pures Appl. 62(3), 383–411 (2017)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Banjai, L., Melenk, J.M., Nochetto, R.H., Otárola, E., Salgado, A.J., Schwab, Ch.: Tensor FEM for spectral fractional diffusion. Found. Comput. Math. (2018). https://doi.org/10.1007/s10208-018-9402-3

  6. 6.

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

    Google Scholar 

  7. 7.

    Bonforte, M., Sire, Y., Vázquez, J.L.: Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains. Discrete Contin. Dyn. Syst. 35(12), 5725–5767 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians II: existence, uniqueness, and qualitative properties of solutions. Trans. Am. Math. Soc. 367(2), 911–941 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

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

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

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

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Caffarelli, L., Stinga, P.R.: Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(3), 767–807 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

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

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    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 

  15. 15.

    Chen, W.: A speculative study of \(2/3\)-order fractional Laplacian modeling of turbulence: some thoughts and conjectures. Chaos 16(2), 1–11 (2006)

    MATH  Google Scholar 

  16. 16.

    Chen, W., Holm, S.: Fractional laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115(4), 1424–1430 (2004)

    Article  Google Scholar 

  17. 17.

    Dahmen, W., Faermann, B., Graham, I.G., Hackbusch, W., Sauter, S.A.: Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method. Math. Comput. 73(247), 1107–1138 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A fractional porous medium equation. Adv. Math. 226(2), 1378–1409 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    de Pablo, A., Quirós, F., Rodríguez, A., Vázquez, J.L.: A general fractional porous medium equation. Commun. Pure Appl. Math. 65(9), 1242–1284 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Diaz, J., Grote, M.J.: Energy conserving explicit local time stepping for second-order wave equations. SIAM J. Sci. Comput. 31(3), 1985–2014 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54(4), 667–696 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

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

  23. 23.

    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)

    Google Scholar 

  24. 24.

    Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  25. 25.

    Fujiwara, D.: Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Jpn. Acad. 43, 82–86 (1967)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Gaspoz, F.D., Heine, C.-J., Siebert, K.G.: Optimal grading of the newest vertex bisection and \(H^1\)-stability of the \(L_2\)-projection. IMA J. Numer. Anal. 36(3), 1217–1241 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Gaspoz, F.D., Morin, P.: Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29(4), 917–936 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Gatto, P., Hesthaven, J.S.: Numerical approximation of the fractional laplacian via \(hp\)-finite elements, with an application to image denoising. J. Sci. Comput. 65(1), 249–270 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

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

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

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

  32. 32.

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

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs. Oxford University Press, New York (1993)

    Google Scholar 

  34. 34.

    Hochbruck, M., Sturm, A.: Error analysis of a second-order locally implicit method for linear Maxwell’s equations. SIAM J. Numer. Anal. 54(5), 3167–3191 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995). Reprint of the 1980 edition

  36. 36.

    Kiselev, A., Nazarov, F., Volberg, A.: Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent. Math. 167(3), 445–453 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Kufner, A.: Weighted Sobolev Spaces, Volume 31 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980). With German, French and Russian summaries

  38. 38.

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

    MathSciNet  MATH  Google Scholar 

  39. 39.

    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

  40. 40.

    Levendorskiĭ, S.Z.: Pricing of the American put under Lévy processes. Int. J. Theor. Appl. Finance 7(3), 303–335 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Lions, J.-L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer, New York (1972). Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181

  42. 42.

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

    Google Scholar 

  43. 43.

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

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Müller, F., Schötzau, D., Schwab, C.: Symmetric interior penalty discontinuous Galerkin methods for elliptic problems in polygons. SIAM J. Numer. Anal. 55(5), 2490–2521 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

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

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    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)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Olver, F.W.J.: Asymptotics and Special Functions. Computer Science and Applied Mathematics. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1974)

    Google Scholar 

  49. 49.

    Otárola, E., Salgado, A.J.: Regularity of solutions to space-time fractional wave equations: a PDE approach. Fract. Calc. Appl. Anal. 21(5), 1262–1293 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Peterseim, D., Schedensack, M.: Relaxing the CFL condition for the wave equation on adaptive meshes. J. Sci. Comput. 72(3), 1196–1213 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Pham, H.: Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optim. 35(2), 145–164 (1997)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Roubíček, T.: Nonlinear Partial Differential Equations with Applications, Volume 153 of International Series of Numerical Mathematics, 2nd edn. Birkhäuser, Basel (2013)

    Google Scholar 

  53. 53.

    Savaré, G.: Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152(1), 176–201 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Schöberl, J.: NETGEN an advancing front 2D/3D-mesh generator based on abstract rules. J. Comput. Vis. Sci. 1, 41–52 (1997)

    MATH  Article  Google Scholar 

  55. 55.

    Schöberl, J.: C++11 implementation of finite elements in NGSolve. Technical report (2014)

  56. 56.

    Silling, S.A.: Why peridynamics? In: Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A. (eds.) Handbook of Peridynamic Modeling, Advances in Applied Mathematics. CRC Press, Boca Raton (2017)

  57. 57.

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

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    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 

  59. 59.

    Turesson, B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces, Volume 1736 of Lecture Notes in Mathematics. Springer, Berlin (2000)

    Google Scholar 

  60. 60.

    Vázquez, J.L., Volzone, B.: Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type. J. Math. Pures Appl. (9) 101(5), 553–582 (2014)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Enrique Otárola.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

EO is partially supported by CONICYT through FONDECYT Project 11180193.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Banjai, L., Otárola, E. A PDE approach to fractional diffusion: a space-fractional wave equation. Numer. Math. 143, 177–222 (2019). https://doi.org/10.1007/s00211-019-01055-5

Download citation

Mathematics Subject Classification

  • 26A33
  • 35J70
  • 35R11
  • 65M12
  • 65M15
  • 65M60