Journal of Scientific Computing

, Volume 72, Issue 2, pp 568–585 | Cite as

DPG Method with Optimal Test Functions for a Fractional Advection Diffusion Equation

  • Vincent J. Ervin
  • Thomas Führer
  • Norbert Heuer
  • Michael Karkulik
Article
  • 203 Downloads

Abstract

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.

Keywords

Fractional diffusion Riemann–Liouville fractional integral DPG method with optimal test functions Ultra-weak formulation 

Mathematics Subject Classification

65N30 

References

  1. 1.
    Babuška, I.: Error-bounds for finite element method. Numer. Math., 16:322–333 (1970/1971)Google Scholar
  2. 2.
    Barrett, J.W., Morton, K.W.: Approximate symmetrization and Petrov–Galerkin methods for diffusion–convection problems. Comput. Methods Appl. Mech. Eng. 45(1–3), 97–122 (1984)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Benson, D.A., Wheatcraft, S., Meerschaert, M.: The fractional-order governing equation of lévy motion. Water Resour. Res. 36(6), 1413–1424 (2000)CrossRefGoogle Scholar
  4. 4.
    Bottasso, C.L., Micheletti, S., Sacco, R.: The discontinuous Petrov–Galerkin method for elliptic problems. Comput. Methods Appl. Mech. Eng. 191(31), 3391–3409 (2002)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bramwell, J., Demkowicz, L., Gopalakrishnan, J., Qiu, W.: A locking-free \(hp\) DPG method for linear elasticity with symmetric stresses. Numer. Math. 122(4), 671–707 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Francaise Automat. Informat. Recherche Opérationnelle Sér. Rouge 8(R–2), 129–151 (1974)MathSciNetMATHGoogle Scholar
  7. 7.
    Broersen, D., Stevenson, R.: A robust Petrov–Galerkin discretisation of convection-diffusion equations. Comput. Math. Appl. 68(11), 1605–1618 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Broersen, D., Stevenson, R.P.: A Petrov–Galerkin discretization with optimal test space of a mild-weak formulation of convection–diffusion equations in mixed form. IMA J. Numer. Anal. 35(1), 39–73 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cessenat, O., Despres, B.: Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 35(1), 255–299 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chan, J., Evans, J.A., Qiu, W.: A dual Petrov–Galerkin finite element method for the convection–diffusion equation. Comput. Math. Appl. 68(11), 1513–1529 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chan, J., Heuer, N., Bui-Thanh, T., Demkowicz, L.: A robust DPG method for convection-dominated diffusion problems II: adjoint boundary conditions and mesh-dependent test norms. Comput. Math. Appl. 67(4), 771–795 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chen, H., Fu, G., Li, J., Qiu, W.: First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems. Comput. Math. Appl. 68(12, part A), 1635–1652 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chen, H., Wang, H.: Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation. J. Comput. Appl. Math. 296, 480–498 (2016)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Chen S., Shen J., Wang L.-L.: Generalized Jacobi functions and their applications to fractional differential equations. Math. Comp. (2016) (to appear)Google Scholar
  15. 15.
    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792–7804 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Demkowicz, L.: Polynomial exact sequences and projection-based interpolation with application to Maxwell equations. Technical Report 06-12, ICES, The University of Texas at Austin (2006)Google Scholar
  17. 17.
    Demkowicz, L., Gopalakrishnan, J.: Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal. 49(5), 1788–1809 (2011)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov–Galerkin methods. Part II: Optimal test functions. Numer. Methods Partial Differ. Equ. 27, 70–105 (2011)CrossRefMATHGoogle Scholar
  19. 19.
    Demkowicz, L., Heuer, N.: Robust DPG method for convection-dominated diffusion problems. SIAM J. Numer. Anal. 51(5), 2514–2537 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Després, B.: Sur une formulation variationnelle de type ultra-faible. C. R. Acad. Sci. Paris Sér. I Math 318(10), 939–944 (1994)MathSciNetMATHGoogle Scholar
  21. 21.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22(3), 558–576 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Feischl, M., Führer, T., Heuer, N., Karkulik, M., Praetorius, D.: Adaptive boundary element methods. Arch. Comput. Methods Eng. 22(3), 309–389 (2015)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Fix, G.J., Roop, J.P.: Least squares finite-element solution of a fractional order two-point boundary value problem. Comput. Math. Appl. 48(7–8), 1017–1033 (2004)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Gopalakrishnan, J., Qiu, W.: An analysis of the practical DPG method. Math. Comp. 83(286), 537–552 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Grisvard, P.: Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985)Google Scholar
  26. 26.
    Heuer, N.: Additive Schwarz method for the \(p\)-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math. 88(3), 485–511 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Heuer, N.: On the equivalence of fractional-order Sobolev semi-norms. J. Math. Anal. Appl. 417(2), 505–518 (2014)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Heuer, N., Karkulik, M.: Discontinuous Petrov–Galerkin boundary elements. Numer. Math. (2016). doi: 10.1007/s00211-016-0824-z
  29. 29.
    Heuer, N., Pinochet, F.: Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions. SIAM J. Numer. Anal. 52(6), 2703–2721 (2014)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows (Papers, Winter Ann. Meeting Amer. Soc. Mech. Engrs., New York, 1979), vol. 34 of AMD, pp. 19–35. Amer. Soc. Mech. Engrs. (ASME), New York (1979)Google Scholar
  31. 31.
    Jin, B., Lazarov, R., Pasciak, J., Rundell, W.: Variational formulation of problems involving fractional order differential operators. Math. Comp. 84(296), 2665–2700 (2015)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Jin, B., Lazarov, R., Zhou, Z.: A Petrov–Galerkin finite element method for factional convection–diffusion equations. SIAM J. Numer. Anal. 54(1), 481–503 (2016)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Li, C., Zeng, F., Liu, F.: Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15(3), 383–406 (2012)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck equation. In: Proceedings of the International Conference on Boundary and Interior Layers—Computational and Asymptotic Methods (BAIL 2002), vol. 166, pp. 209–219 (2004)Google Scholar
  35. 35.
    Liu, Q., Liu, F., Turner, I., Anh, V.: Finite element approximation for a modified anomalous subdiffusion equation. Appl. Math. Model. 35(8), 4103–4116 (2011)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. In: Fractals and fractional calculus in continuum mechanics (Udine. 1996), vol. 378 of CISM Courses and Lectures, pp. 291–348. Springer, Vienna (1997)Google Scholar
  37. 37.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Rudin, W.: Functional Analysis. International Series in Pure and Applied Mathematics, 2nd edn. McGraw-Hill Inc., New York (1991)Google Scholar
  39. 39.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives. In: Theory and applications, edited and with a foreword by Nikol’skiĭ, S.M., Translated from the 1987 Russian original, Revised by the authors. Gordon and Breach Science Publishers, Yverdon (1993)Google Scholar
  40. 40.
    Shlesinger, M.F., West, B.J., Klafter, J.: Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58(11), 1100–1103 (1987)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220(2), 813–823 (2007)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Wang, H., Basu, T.S.: A fast finite difference method for two-dimensional space-fractional diffusion equations. SIAM J. Sci. Comput. 34(5), A2444–A2458 (2012)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Wang, H., Yang, D.: Wellposedness of variable-coefficient conservative fractional elliptic differential equations. SIAM J. Numer. Anal. 51(2), 1088–1107 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Wang, H., Yang, D., Zhu, S.: Inhomogeneous Dirichlet boundary-value problems of space-fractional diffusion equations and their finite element approximations. SIAM J. Numer. Anal. 52(3), 1292–1310 (2014)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Wang, H., Yang, D., Zhu, S.: A Petrov–Galerkin finite element method for variable-coefficient fractional diffusion equations. Comput. Methods Appl. Mech. Eng. 290, 45–56 (2015)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Wang, H., Zhang, X.: A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations. J. Comput. Phys. 281, 67–81 (2015)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Xu, J., Zikatanov, L.: Some observations on Babuška and Brezzi theories. Numer. Math. 94(1), 195–202 (2003)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Xu, Q., Hesthaven, J.S.: Discontinuous Galerkin method for fractional convection–diffusion equations. SIAM J. Numer. Anal. 52(1), 405–423 (2014)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Zaslavsky, G.M., Stevens, D., Weitzner, H.: Self-similar transport in incomplete chaos. Phys. Rev. E 48(3), 1683–1694 (1993)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Zayernouri, M., Ainsworth, M., Karniadakis, G.E.: A unified Petrov–Galerkin spectral method for fractional PDEs. Comput. Methods Appl. Mech. Eng. 283, 1545–1569 (2015)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D., Calo, V.M.: A class of discontinuous Petrov–Galerkin methods. Part IV: the optimal test norm and time-harmonic wave propagation in 1D. J. Comput. Phys. 230(7), 2406–2432 (2011)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Vincent J. Ervin
    • 1
  • Thomas Führer
    • 2
  • Norbert Heuer
    • 2
  • Michael Karkulik
    • 3
  1. 1.Department of Mathematical SciencesClemson UniversityClemsonUSA
  2. 2.Facultad de MatemáticasPontificia Universidad Católica de ChileMaculChile
  3. 3.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile

Personalised recommendations