Numerische Mathematik

, 107:257 | Cite as

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



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 


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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

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