Abstract
SummaryEfficient and robust iterative methods are developed for solving the linear systems of equations arising from stochastic finite element methods for single phase fluid flow in porous media. Permeability is assumed to vary randomly in space according to some given correlation function. In the companion paper, herein referred to as Part 1, permeability was approximated using a truncated Karhunen‐Loève expansion (KLE). The stochastic variability of permeability is modeled using lognormal random fields and the truncated KLE is projected onto a polynomial chaos basis. This results in a stochastic nonlinear problem since the random fields are represented using polynomial chaos containing terms that are generally nonlinear in the random variables. Symmetric block Gauss‐Seidel used as a preconditioner for CG is shown to be efficient and robust for stochastic finite element method.
Highlights
This article is concerned with the numerical simulation of flow through porous media
For the stochastically nonlinear case this situation is exacerbated by the fact that every block of the global system has non-zero entries. To overcome this important limitation, we propose an alternative preconditioner for stochastic finite element method (SFEM) in which the off-diagonal blocks of the global system are included in the preconditioned system using a block symmetric Gauss-Seidel algorithm
For test problem 3, in which the conductivity field is discontinuous, the predicted CPU times on a mesh with h = 1∕64 with pu = 4 are 240 and 2120 seconds, respectively, for d = 4 and d = 6. If these times are compared with the CPU times for Monte Carlo Method (MCM) on the much simpler test problem 1 given in Table 1, we see that the stochastic mixed finite element method (SMFEM) with block GS preconditioner is at least an order of magnitude faster in terms of CPU time
Summary
This article is concerned with the numerical simulation of flow through porous media. The second choice is based on the polynomial chaos (PC) method as outlined in the original work of Ghanem and Spanos.[5,6] In this approach, spectral representations of uncertainty in terms of multi-dimensional Hermite polynomials or polynomial chaos expansions are used to approximate both the model parameters and the solution This enables the stochastic equations to be replaced by deterministic systems of PDEs which are truncated and discretized. This approach is TRAVERSO and PHILLIPS efficient for those cases in which the off-diagonal blocks of A hold significant information on the permeability, that is, problems with large values of σ It is evident from the literature and from our computational analysis that mean-based preconditioners cannot be robust with respect to the permeability since they only include, in the preconditioned system, information associated with the mean value of the spatial random field.
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