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, 65C30Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Wiener chaos, stochastic PDE, stochastic modeling, stochastic finite element method. *Corresponding author. Email: xlwan@math.lsu.edu (X. Wan) |