Numerische Mathematik

, 107:257 | Cite as

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Article

Abstract

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.

Mathematics Subject Classification (1991)

60H15 60H35 65C30 65N30 65N35 65N12 

References

  1. 1.
    Babuška, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. (to appear)Google Scholar
  2. 2.
    Babuška I., Tempone R., Zouraris G.E. (2004). Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42: 800–825 CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bertoluzza S. (1997). Interior estimates for the wavelet Galerkin method. Numer. Math. 78: 1–20 MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Brezzi F., Fortin M. (1991). Mixed and Hybrid Finite Element Methods. Springer, New York MATHGoogle Scholar
  5. 5.
    Cameron R., Martin W. (1947). The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann. Math. 48: 385–392 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Caflisch, R.E.: Monte Carlo and quasi-Monte Carlo methods. Acta Numer., 1–49 (1998)Google Scholar
  7. 7.
    Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A. (2006). Spectral Methods. Fundamentals in Single Domains. Springer, Berlin MATHGoogle Scholar
  8. 8.
    Deb M.K., Babuška I., Oden J.T. (2001). Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput. Methods Appl. Mech. Eng. 190: 6359–6372 MATHCrossRefGoogle Scholar
  9. 9.
    Ditzian Z., Totik V. (1987). Moduli of Smoothness. Springer, New York MATHGoogle Scholar
  10. 10.
    Ghanem R., Spanos P.D. (1991). Stochastic Finite Elements—A Spectral Approach. Springer, Berlin MATHGoogle Scholar
  11. 11.
    Gerstner T., Griebel M. (1998). Numerical integration using sparse grids. Numer. Algorithms 18: 209–232 MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Girault V., Glowinski R. (1995). Error analysis of a fictitious domain method applied to a Dirichlet problem. Japan J. Ind. Appl. Math. 12: 487–514 MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Glowinski R., Pan T.S., Périaux J. (1993). A least squares/fictitious domain method for mixed problems and Neumann problems. In: Lions, J.L., Baiocchi, C. (eds) Boundary Value Problems for Partial Differential Equations and Applications, pp 159–178. Masson, Paris Google Scholar
  14. 14.
    Glowinski R., Pan T., Periaux J. (1994). A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng. 111: 283–303 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Haslinger J., Kozubek T. (2000). A fictitious domain approach for a class of Neumann boundary value problems with applications in shape optimization. East-West J Numer. Math. 8: 1–26 MATHMathSciNetGoogle Scholar
  16. 16.
    Haslinger, J., Kozubek, T., Kucera, R., Peichl, G.: Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach. Numer. Linear Algebra Appl. (2007) (to appear)Google Scholar
  17. 17.
    Haslinger J., Kozubek T., Kunisch K., Peichl G. (2003). Shape optimization and fictitious domain approach for solving free boundary problems of Bernoulli type. COAP 26: 231–251 MATHMathSciNetGoogle Scholar
  18. 18.
    Haslinger J., Mäkinen R.A.E. (2003). Introduction to Shape Optimization, Theory, Approximation and Computation. SIAM, Philadelphia MATHGoogle Scholar
  19. 19.
    Hosden, S., Walters, R.W., Perez, R.: A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. AIAA Paper 2006-0891Google Scholar
  20. 20.
    Kleiber M., Hien T.D. (1992). The Stochastic Finite Element Method. Basic Perturbation Technique and Computer Implementation. Wiley, Chichester MATHGoogle Scholar
  21. 21.
    Kucera, R., Kozubek, T., Haslinger, J.: On solving non-symmetric saddle-point systems arising from fictitious domain approaches. In: Proceedings of PANM06, Prague, Czech Republic, pp. 165–171 (2006)Google Scholar
  22. 22.
    Loève M. (1977). Probability Theory. Springer-Verlag, Berlin MATHGoogle Scholar
  23. 23.
    Mastroianni G., Monegato G. (2006). Truncated approximation processes on unbounded intervals and their applications. Rend. Circ. Mat. Palermo 55: 123–139 CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Mathelin I., Hussaini M.Y., Zang T.A. (2005). Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38: 209–236 MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Matthies, H.G., Keese, A.: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, pp. 1295–1331 in [28]Google Scholar
  26. 26.
    Mommer M.S. (2006). A smoothness preserving fictitious domain method for elliptic boundary-value problems. IMA J. Numer. Anal. 26: 503–524 MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Oksendal B. (1998). Stochastic Differential Equations—An Introduction with Applications. Springer, Berlin Google Scholar
  28. 28.
    Schuëller, G.I. (ed.): Computational Methods in Stochastic Mechanics and Reliability Analysis. Special Issue 12–16, Comput. Methods Appl. Mech. Eng. 194 (2005)Google Scholar
  29. 29.
    Schwab Ch., Todor R.A. (2003). Sparse finite elements for stochastic elliptic problems-higher order moments. Computing 71: 43–63 MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Todor, R.A., Schwab, Ch.: Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. Research Rep. No. 2006-05, Seminar für Angewandte Mathematik, ETH ZurichGoogle Scholar
  31. 31.
    Walters, R.W.: Towards stochastic fluid mechanics via polynomial chaos. AIAA Paper 2003-0413Google Scholar
  32. 32.
    Wiener N. (1938). The homogeneous chaos. Am. J. Math. 60: 897–936 MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Xiu D., Karniadakis G.E. (2002). The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24: 619–644 MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Xiu D., Karniadakis G.E. (2003). Modeling uncertainty in flow simulation via generalized polynomial chaos. J. Comput. Phys. 187: 137–167 MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Xiu D., Tartakovsky D.M. (2006). Numerical methods for differential equations in random domains. SIAM J. Sci. Comput. 28: 1167–1185 MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly
  2. 2.Department of Applied MathematicsVSB-Technical University of OstravaOstravaCzech Republik

Personalised recommendations