Preprint A587/2008
Aproximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity
Marcus Sarkis | Galvis, Juan Carlos
Keywords: White noise analysis; Wiener-Chaos expansions; finite elements;
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 {\it 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.