A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.

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Journal of Computational Physics
Efficient numerical methods for deformable porous media. Application to carbon dioxide storage
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Kumar, P., Luo, P., Gaspar, F., & Oosterlee, K. (2018). A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system. Journal of Computational Physics, 371, 382–408. doi:10.1016/j.jcp.2018.05.046