The Stochastic Galerkin Method for Darcy Flow Problem with Log-Normal Random Field Coefficients

Michal Beres, Simona Domesova

The Stochastic Galerkin Method for Darcy Flow Problem with Log-Normal Random Field Coefficients

Číslo: 2/2017
Periodikum: Advances in Electrical and Electronic Engineering
DOI: 10.15598/aeee.v15i2.2280

Klíčová slova: Darcy flow; Gaussian random field; Karhunen-Loeve decomposition; polynomial chaos; Stochastic Galerkin method, Gaussovo náhodné pole; Karhunen-Loeve rozklad; Polynomický chaos; Stochastická metoda Galerkina.

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Anotace: This article presents a study of the Stochastic Galerkin Method (SGM) applied to the Darcy flow problem with a log-normally distributed random material field given by a mean value and an autocovariance function. We divide the solution of the problem into two parts. The first one is the decomposition of a random field into a sum of products of a random vector and a function of spatial coordinates; this can be achieved using the Karhunen-Loeve expansion. The second part is the solution of the problem using SGM. SGM is a simple extension of the Galerkin method in which the random variables represent additional problem dimensions. For the discretization of the problem, we use a finite element basis for spatial variables and a polynomial chaos discretization for random variables. The results of SGM can be utilised for the analysis of the problem, such as the examination of the average flow, or as a tool for the Bayesian approach to inverse problems.