A fictitious domain approach to the numerical solution of PDEs in stochastic domains |
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Authors: | Claudio Canuto Tomas Kozubek |
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Institution: | (1) Dipartimento di Matematica, Politecnico di Torino, 10129 Torino, Italy;(2) Department of Applied Mathematics, VSB-Technical University of Ostrava, 70833 Ostrava, Czech Republik |
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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. |
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Keywords: | 60H15 60H35 65C30 65N30 65N35 65N12 |
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