Document Type

Article

Publication Date

10-1-2009

Publication Title

SIAM Journal on Numerical Analysis

Abstract

We consider a stochastic Darcy's pressure equation whose coefficient is generated by a white noise process on a Hilbert space employing the ordinary (rather than the Wick) product. A weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf-sup condition and well-posedness of the problem. We generalize the numerical approximations proposed in Benth and Theting [Stochastic Anal. Appl., 20 (2002), pp. 1191-1223] for Wick stochastic partial differential equations to the ordinary product stochastic pressure equation. We establish discrete inf-sup conditions and provide a priori error estimates for a wide class of norms. The proposed numerical approximation is based on Wiener-Chaos finite element methods and yields a positive definite symmetric linear system. We also improve and generalize the approximation results of Benth and Gjerde [Stochastics Stochastics Rep., 63 (1998), pp. 313-326] and Cao [Stochastics, 78 (2006), pp. 179-187] when a (generalized) process is truncated by a finite Wiener-Chaos expansion. Finally, we present numerical experiments to validate the results.

Volume

47

Issue

5

First Page Number

3624

Last Page Number

3651

DOI

10.1137/080717924

Publisher Statement

© 2009, SIAM Publications.

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Mathematics Commons

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