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July  2006, 6(4): 941-955. doi: 10.3934/dcdsb.2006.6.941

Stochastic Galerkin method for elliptic spdes: A white noise approach

Luis J. Roman 1, and  Marcus Sarkis 1,

1. 

Worcester Polytechnic Institute, Department of Mathematical Sciences, 100 Institute Rd, Worcester, MA 01609-2280, United States, United States

Received  January 2005 Revised  October 2005 Published  April 2006

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 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.
Keywords: Stochastic Differential Equations, White Noise, Galerkin Method, Elliptic Partial Differential Equations..
Mathematics Subject Classification: Primary: 60H30, 60H35, 65C20; Secondary: 65C30, 65N3.
Citation: Luis J. Roman, Marcus Sarkis. Stochastic Galerkin method for elliptic spdes: A white noise approach. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 941-955. doi: 10.3934/dcdsb.2006.6.941
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