Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods

Ivan Graham, Robert Scheichl, Elisabeth Ullmann

Research output: Contribution to journalArticlepeer-review

17 Citations (SciVal)
197 Downloads (Pure)


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 mass-conservative 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 non-trivial 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 languageEnglish
Pages (from-to)41-75
Number of pages35
JournalStochastic Partial Differential Equations : Analysis and Computations
Issue number1
Early online date12 Jun 2015
Publication statusPublished - 1 Mar 2016


  • Multilevel Monte Carlo
  • random porous media
  • mixed finite elements
  • lognormal coefficients


Dive into the research topics of 'Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods'. Together they form a unique fingerprint.

Cite this