Advertisement

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

  • 267 Accesses

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.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

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)

  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)

  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)

  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)

  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)

  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)

  10. 10.

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

  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)

  12. 12.

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

  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)

  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)

  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)

  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)

  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)

  18. 18.

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

  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)

  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)

  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)

  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)

  24. 24.

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

  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)

  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)

  27. 27.

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

  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)

  29. 29.

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

  30. 30.

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

  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)

  33. 33.

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

  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)

  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)

  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)

  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)

  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)

  43. 43.

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

  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)

  45. 45.

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

  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)

  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)

  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)

  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)

  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)

  51. 51.

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

  52. 52.

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

  53. 53.

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

  54. 54.

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

  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)

  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)

  59. 59.

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

  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)

Download references

Author information

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