Preprint A541/2007
Stochastic Galerkin method for elliptic SPDEs: A white noise
approach

Marcus Sarkis | Roman , Luis

**Keywords: **
Stochastic Differential Equations | Galerkin Method | White
Noise | Elliptic Partial Differential Equations

An equation that arises in mathematical studies of the
transport of pollutants in groundwater and of oil recovery
processes is of the form:
$-\nabla_{x}\cdot(\kappa(x,\cdot)\nabla_{x}u(x,\omega))=f(x)$, for
$x\in D$, where $\kappa(x,\cdot)$, the permeability tensor, is
random and models the properties of the rocks, which are not know
with certainty. Further, geostatistical models assume
$\kappa(x,\cdot)$ to be a log-normal random field. The use of
Monte Carlo methods to approximate the expected value of
$u(x,\cdot)$, higher moments, or other functionals of
$u(x,\cdot)$, require solving similar system of equations many
times as trajectories are considered, thus it becomes expensive
and impractical. In this paper, we present and explain several
advantages of using the {\it White Noise} probability space as a
natural framework for this problem. Applying properly and timely
the Wiener-Itô Chaos
decomposition and an eigenspace decomposition, we obtain
a symmetric positive definite linear system of equations
whose solutions are the coefficients of a Galerkin-type
approximation to the solution of the original equation. Moreover,
this approach reduces the simulation of the approximation to
$u(x,\omega)$ for a fixed $\omega$, to the simulation of a finite
number of independent normally distributed random variables.