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Abstract
This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here massconservative lowest order Raviart–Thomas mixed finite elements. This is very important since local mass conservation is highly desirable in realistic groundwater flow problems. Due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the operator the analysis is nontrivial in the presence of lognormal random fields. We establish finite element error bounds for Darcy velocity and pressure, as well as for a more accurate recovered pressure approximation. We then apply the error bounds to prove convergence of the multilevel Monte Carlo algorithm for estimating statistics of these quantities. Moreover, we prove convergence for a class of bounded, linear functionals of the Darcy velocity. An important special case is the approximation of the effective permeability in a 2D flow cell. We perform numerical experiments to confirm the convergence results.
Original language  English 

Pages (fromto)  4175 
Number of pages  35 
Journal  Stochastic Partial Differential Equations : Analysis and Computations 
Volume  4 
Issue number  1 
Early online date  12 Jun 2015 
DOIs  
Publication status  Published  1 Mar 2016 
Keywords
 Multilevel Monte Carlo
 random porous media
 mixed finite elements
 lognormal coefficients
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Dive into the research topics of 'Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods'. Together they form a unique fingerprint.Projects
 2 Finished

Multilevel Monte Carlo Methods for Elliptic Problems
Scheichl, R.
Engineering and Physical Sciences Research Council
1/07/11 → 30/06/14
Project: Research council

Adaptive Multiscale Methods for Approximation and Preconditioning
Engineering and Physical Sciences Research Council
1/05/10 → 31/07/11
Project: Research council