Aproximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity
Juan Carlos Galvis ,
Marcus Sarkis
Keywords:
White noise analysis; Wiener-Chaos expansions; finite elements;
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 {\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.
MSC 2000:
60H15 Stochastic partial differential equations
60H30 Applications of stochastic analysis (to PDE, etc.)
Notes:
To appear in SIAM Journal of Numerical Analysis
