Numerische Mathematik

, Volume 141, Issue 4, pp 1043–1077 | Cite as

Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise

  • Max Gunzburger
  • Buyang Li
  • Jilu WangEmail author


Numerical approximation of a stochastic partial integro-differential equation driven by a space-time white noise is studied by truncating a series representation of the noise, with finite element method for spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.



  1. 1.
    Allen, E.J., Novosel, S.J., Zhang, Z.: Finite element and difference approximation of some linear stochastic partial differential equations. Stochastics and Stochastics Reports, 64 (1998)Google Scholar
  2. 2.
    Anton, R., Cohen, D., Larsson, S., Wang, X.: Full discretization of semilinear stochastic wave equations driven by multiplicative noise. SIAM J. Numer. Anal. 54(2), 1093–1119 (2016)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arendt, W., Batty, C.J., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, 2nd edn. Birkhäuser, Basel (2011)zbMATHGoogle Scholar
  4. 4.
    Baňas, L., Brzeźnik, Z., Prohl, A.: Computational studies for the stochastic Landau–Lifshitz–Gilbert equation. SIAM J. Sci. Comput. 35(1), B62–B81 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)zbMATHGoogle Scholar
  6. 6.
    Clément, P., Da Prato, G.: Some results on stochastic convolutions arising in Volterra equations perturbed by noise. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7(3), 147–153 (1996)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cuesta, E., Lubich, C., Palencia, C.: Convolution quadrature time discretization of fractional diffusion-wave equations. Math. Comput. 75(254), 673–696 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, 2nd edn. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  9. 9.
    Du, Q., Zhang, T.: Numerical approximation of some linear stochastic partial differential equations driven by special additive noises. SIAM J. Numer. Anal. 40(4), 1421–1445 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Feng, X., Li, Y., Prohl, A.: Finite element approximations of the stochastic mean curvature flow of planar curves of graphs. Stoch. PDE Anal. Comput. 2(1), 54–83 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics. Springer, New York (2008)zbMATHGoogle Scholar
  12. 12.
    Gunzburger, M., Li, B., Wang, J.: Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise. Math. Comput. (2019). Google Scholar
  13. 13.
    Gunzburger, M., Wang, J.: A second-order Crank-Nicolson method for time-fractional PDEs. Int. J. Numer. Anal. Model. 16(2), 225–239 (2019)MathSciNetzbMATHGoogle Scholar
  14. 14.
    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)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39(6), A3129–A3152 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Jin, B., Li, B., Zhou, Z.: An analysis of the Crank–Nicolson method for subdiffusion. IMA J. Numer. Anal. 38(1), 518–541 (2018)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam (2006)zbMATHGoogle Scholar
  18. 18.
    Kovács, M., Printems, J.: Strong order of convergence of a fully discrete approximation of linear stochastic Volterra type evolution equation. Math. Comput. 83(289), 2325–2346 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Laptev, A.: Dirichlet and Neumann eigenvalue problems on domains in Euclidean spaces. J. Funct. Anal. 151, 531–545 (1997)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Li, P., Yau, S.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88(3), 309–318 (1983)zbMATHGoogle Scholar
  21. 21.
    Lubich, C.: Convolution quadrature and discretized operational calculus. I. Numer. Math. 52(2), 129–145 (1988)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lubich, C.: Convolution quadrature and discretized operational calculus. II. Numer. Math. 52(4), 413–425 (1988)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lubich, C., Sloan, I.H., Thomée, V.: Nonsmooth data error estimates for approximations of an evolution equation with a positive-type memory term. Math. Comput. 65(213), 1–17 (1996)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mijena, J.B., Nane, E.: Space-time fractional stochastic partial differential equations. Stoch. Process. Appl. 125, 3301–3326 (2015)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sanz-Serna, J.M.: A numerical method for a partial integro-differential equation. SIAM J. Numer. Anal. 25(2), 319–327 (1988)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Shardlow, T.: Numerical methods for stochastic parabolic PDEs. Numer. Funct. Anal. Optim. 20, 121–145 (1999)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Strauss, W.A.: Partial Differential Equations: An Introduction. Wiley, New York (2008)zbMATHGoogle Scholar
  28. 28.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  29. 29.
    Vázquez, L., Velasco, M., Vázquez-Poletti, J., Llorente, M., Usero, D., Jiménez, S.: Modeling and simulation of the atmospheric dust dynamic: fractional calculus and cloud computing. Int. J. Numer. Anal. Model. 15(1), 74–85 (2018)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Yan, Y.: Galerkin finite element methods for stochastic parabolic partial differential equations. SIAM J. Numer. Anal. 43(4), 1363–1384 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Scientific ComputingFlorida State UniversityTallahasseeUSA
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloonHong Kong
  3. 3.Department of Mathematics and StatisticsMississippi State UniversityMississippi StateUSA

Personalised recommendations