Commun. Comput. Phys., 11 (2012), pp. 775-796.


A Discussion on Two Stochastic Elliptic Modeling Strategies

Xiaoliang Wan 1*

1 Department of Mathematics and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA 70803, USA.

Received 30 June 2010; Accepted (in revised version) 14 April 2011
Available online 28 October 2011
doi:10.4208/cicp.300610.140411a

Abstract

Based on the study of two commonly used stochastic elliptic models: $I: -\nabla\cdot(a(\bx,\omega)\cdot\nabla u(\bx,\omega))=f(\bx)$ and $II: -\nabla\cdot(a(\bx,\omega)\diamond\nabla u(\bx,\omega))=f(\bx)$, we constructed a new stochastic elliptic model III: $-\nabla\cdot ((a^{-1})^{\diamond(-1)}\diamond\nabla u(\bx,\omega))=f(\bx)$, In [20]. The difference between models I and II is twofold: a scaling factor induced by the way of applying the Wick product and the regularization induced by the Wick product itself. In [20], we showed that model III has the same scaling factor as model I. In this paper we present a detailed discussion about the difference between models I and III with respect to the two characteristic parameters of the random coefficient, i.e., the standard deviation $\sigma$ and the correlation length $l_c$. Numerical results are presented for both one- and two-dimensional cases.

AMS subject classifications: 60H15, 65C20, 65C30

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Key words: Wiener chaos, stochastic PDE, stochastic modeling, stochastic finite element method.

*Corresponding author.
Email: xlwan@math.lsu.edu (X. Wan)
 

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