Robust Preconditioning for Second-Order Elliptic PDEs with Random Field Coefficients

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Abstract

Fluid flow and the transport of chemicals in flows in heterogeneous porous media are modelled mathematically using partial differential equations (PDEs). In deterministic modelling, material properties of the porous medium are assumed to be known explicitly. This assumption leads to tractable computations. To tackle the more realistic stochastic groundwater flow problem, it is necessary to represent the unknown permeability coefficients as random fields with prescribed statistical properties. Traditionally, large numbers of deterministic problems are solved in a Monte Carlo framework and the solutions averaged to obtain statistical properties of the solution variables. Alternatively, the so-called stochastic finite element method (SFEM) discretises the probabilistic dimension of the PDE directly. However, this approach has not gained popularity with practitioners due to a perceived high computational cost. In this report we solve the stochastic Darcy flow problem via traditional and stochastic finite element techniques, in primal and mixed formulation where appropriate. Permeability coefficients are represented using Gaussian or lognormal random fields. We focus on fast and efficient linear algebra techniques for solving both the large numbers of deterministic problems required by the Monte Carlo approach and, in contrast, the single, structured, but extremely large linear system that arises as a consequence of the SFEM. To achieve optimal computational complexity, black-box algebraic multigrid is exploited in the design of fast solvers.

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