Numerische Mathematik

, Volume 95, Issue 4, pp 707–734

Sparse finite elements for elliptic problems with stochastic loading

Article

Summary.

We formulate elliptic boundary value problems with stochastic loading in a bounded domain D⊂ℝd. We show well-posedness of the problem in stochastic Sobolev spaces and we derive a deterministic elliptic PDE in D×D for the spatial correlation of the random solution. We show well-posedness and regularity results for this PDE in a scale of weighted Sobolev spaces with mixed highest order derivatives. Discretization with sparse tensor products of any hierarchic finite element (FE) spaces in D yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data. Multilevel preconditioning in D×D allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation.

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References

  1. 1.
    Babuška, I.: On Randomised Solutions of Laplace’s Equation. Časopis pro Pěstováni Matematiky 86, 269–275 (1961)Google Scholar
  2. 2.
    Babuška, I., Guo, B.Q.: Regularity of the solution of elliptic problems with piecewise analytic data. Part I. Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19, 172–203 (1988)Google Scholar
  3. 3.
    Dahmen, W., Kunoth, A., Urban, K.: Biorthogonal Spline-Wavelets on the Interval – Stability and Moment Conditions. Appl. Comp. Harm. Anal. 6, 132–196 (1999)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Dahmen, W., Stevenson, R.: Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions. SIAM J. Numer. Anal. 37, 319–352 (1999)MathSciNetMATHGoogle Scholar
  5. 5.
    Deb, M.K., Babuška, I., Oden, J.T.: Solution of Stochastic Partial Differential Equations Using Galerkin Finite Element Techniques. Preprint, University of Texas at Austin, 2001Google Scholar
  6. 6.
    Dettinger, M.: Numerical Modeling of Aquifer Systems Under Uncertainty: A Second Moment Analysis, MSc. Thesis, MIT, Department of Civil Engineering, 1979Google Scholar
  7. 7.
    Ghanem, R.G., Spanos, P.D.: Stochastic finite elements: a spectral approach, Springer-Verlag, 1991Google Scholar
  8. 8.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977Google Scholar
  9. 9.
    Golub, G., Van Loan, C.F.: Matrix Computations, 4th edition, Johns Hopkins University Press, 1996Google Scholar
  10. 10.
    Griebel, M., Oswald, P.: Tensor product type subspace splittings and multilevel iterative methods for anisotropic problems. Adv. Comput. Math. 4, 171–206 (1999)MATHGoogle Scholar
  11. 11.
    Griebel, M., Oswald, P., Schiekofer, T.: Sparse grids for boundary integral equations. Numer. Mathematik 83, 279–312 (1999)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Holden, H., Oksendal, B., Uboe, J., Zhang, T.: Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Birkhäuser, 1996Google Scholar
  13. 13.
    Kampé de Fériet, J.: Random Solutions of Partial Differential Equations, Proc. of the Third Berkeley Symp. on Math. Statistics and Probability, III 199–208Google Scholar
  14. 14.
    Keller, J.B.: Stochastic Equations and Wave Propagation in Random Media, Bellman, 1964Google Scholar
  15. 15.
    Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. John Wiley & Sons, 1992Google Scholar
  16. 16.
    Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. 3rd edition, Springer-Verlag, 1999Google Scholar
  17. 17.
    Matache, A.M.: Sparse Two Scale FEM for Homogenization Problems. J. Sc. Comp. 17, 659–669 (2002)CrossRefMATHGoogle Scholar
  18. 18.
    Oksendal, B.: Stochastic differential equations: an introduction with applications, 3rd edition. Springer-Verlag, 1992Google Scholar
  19. 19.
    Protter, P.: Stochastic integration and differential equations: a new approach, 3rd edition, Springer-Verlag, 1995Google Scholar
  20. 20.
    Strichartz, R.S.: A guide to distribution theory and Fourier transforms, CRC Press, Boca Raton, 1994Google Scholar
  21. 21.
    Todor, R.A.: Doctoral Dissertation ETHZ. In preparationGoogle Scholar
  22. 22.
    Yaglom, A.M.: An Introduction to the Theory of Stationary Random Functions. Prentice-Hall, Englewood Cliffs, New Jersey, 1962Google Scholar
  23. 23.
    Yosida, K.: Functional Analysis, Springer-Verlag, 1964Google Scholar
  24. 24.
    Zenger, Ch.: Sparse Grids, Parallel Algorithms for PDE’s-Proceedings of the 6th GAMM-Seminar, Kiel, 1990Google Scholar

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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Seminar for Applied MathematicsZürichSwitzerland

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