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Numerische Mathematik

, Volume 106, Issue 2, pp 181–198 | Cite as

Finite element methods for semilinear elliptic stochastic partial differential equations

  • Yanzhao Cao
  • Hongtao Yang
  • Li Yin
Article

Abstract

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results.

Mathematics Subject Classification (2000)

65N30 65N15 65C30 60H15 

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References

  1. 1.
    Adams R.A. (1975). Sobolev Spaces. Academic, New York zbMATHGoogle Scholar
  2. 2.
    Allen E.J., Novosel S.J. and Zhang Z. (1998). Finite element and difference approximation of some linear stochastic partial differential equations. Stochastic Stochastic Rep. 64: 117–142 zbMATHMathSciNetGoogle Scholar
  3. 3.
    Babuska I., Tempone R. and Zouraris G. (2004). Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42: 800–825 zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Benth F.E. and Gjerde J. (1998). Convergence rates for finite element approximations of stochastic partial differential equations. Stochastics Stochastics Rep. 63: 313–326 zbMATHMathSciNetGoogle Scholar
  5. 5.
    Brenner S. and Scott L.R. (1994). The Mathematical Theory of Finite Element Methods. Springer, Heidelberg zbMATHGoogle Scholar
  6. 6.
    Buckdahn, R., Pardoux, E.: Monotonicity methods for white noise driven quasi-linear PDEs. Diffusion processes and related problems in analysis, vol. 1 (Evaston IL, 1989), Progr. Probab. 22, Birkhauser, Boston, 1990, pp. 219–233Google Scholar
  7. 7.
    Cao Y. (2006). On Convergence Rate of Wiener-Ito Expansion for generalized random variables. Stochastics 78(3): 179–187 zbMATHMathSciNetGoogle Scholar
  8. 8.
    Courant R. and Hilbert D. (1953). Methods of Mathematical Physics, vol. 1. Interscience, New York Google Scholar
  9. 9.
    Du Q. and Tianyu Z. (2002). Numerical approximation of some linear stochastic partial differential equations driven by special additive noise. SIAM J. Numer. Anal. 4: 1421–1445 CrossRefGoogle Scholar
  10. 10.
    Grisvard P. (1985). Elliptic Problems in Nonsmooth Domains. Pitman Advanced Pub. Program, Boston zbMATHGoogle Scholar
  11. 11.
    Gyöngy I. (1999). Lattice approximations for stochastic quasi-linear parabolic partial differential equations driven by space-time white noise II. Potential Anal. 11: 1–37 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Gyöngy I. and Martinez T. (2006). On the approximation of solutions of stochastic partial differential equations of elliptic type. Stochastics 78(4): 213–231 zbMATHMathSciNetGoogle Scholar
  13. 13.
    Hausenblas E. (2003). Approximation for semilinear stochastic evolution equations. Potential Anal. 18: 141–186 zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hou T., Kim H., Rozovskii B. and Zhou H. (2004). Wiener chaos expansions and numerical solutions randomly forced equations of fluid mechanics. Inter. J. Comp. Math. Appl. 4: 1–14 Google Scholar
  15. 15.
    Keese A., Matthies H. (2003) Parallel computation of stochastic groundwater flow. NIC Symposium 2004, Proceedings, John von Neumann Institute for Computing, Julich, NIC series 20, 399–408Google Scholar
  16. 16.
    KloedenEckhard Platen P.E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg Google Scholar
  17. 17.
    Le Maitre O., Knio O., Najm H. and Ghanem R. (2001). A stochastic projection method for fluid flow: Basic formulation. J. Comp. Phys. 173: 481–511 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lions J.L. (1969). The Method of Quasi-reversibility; applications to Partial Differential Equations. American Elsevier Pub. Co., New York Google Scholar
  19. 19.
    Shardlow, T.: Modified Equation for Stochastic Differential Equation. MCCM preprint No. 455, 2004Google Scholar
  20. 20.
    Shardlow T. (2003). Weak Convergence of A Numerical Method for A Stochastic Heat Equation. BIT Numer. Math. 43: 179–193 zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Thething T.G. (2000). Solving Wick-stochastic boundary value problem using a finite element method. Stochastics Stochastics Rep. 70: 241–270 MathSciNetGoogle Scholar
  22. 22.
    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1180, Springer, Heidelberg, pp. 265–439 (1986)Google Scholar
  23. 23.
    Xiu D. and Karniadakis G.E. (2003). Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comp. Phys. 187: 137–167 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Yan, Y.: The Finite Element Method for a Linear Stochastic Parabolic Partial Differential Equation Driven by Additive Noise. Chalmers Finite Element Center, Chalmers University of Technology, Preprint #7 (2003)Google Scholar
  25. 25.
    Yan, Y.: A Finite Element Method for a Nonlinear Stochastic Parabolic Equation. Chalmers Finite Element Center, Chalmers University of Technology, Preprint #8 (2003)Google Scholar
  26. 26.
    Zeidler E. (1985). Nonlinear functional analysis and its applications II/B: nonlinear monotone operators. Springer, New York Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of mathematicsFlorida A&M UniversityTallahasseeUSA
  2. 2.Department of MathematicsUniversity of LouisianaLafayetteUSA
  3. 3.Department of mathematicsJilin UniversityChangchunChina

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