Numerische Mathematik

, Volume 95, Issue 4, pp 707–734

Sparse finite elements for elliptic problems with stochastic loading

Authors

    • Seminar for Applied Mathematics
  • Radu-Alexandru Todor
    • Seminar for Applied Mathematics
Article

DOI: 10.1007/s00211-003-0455-z

Cite this article as:
Schwab, C. & Todor, R. Numer. Math. (2003) 95: 707. doi:10.1007/s00211-003-0455-z

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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2003