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.

Contaminant transport, Darcy–Stokes flow, MLMC, Multigrid, UQ, Uzawa smoother
Journal of Computational Physics
Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Kumar, P, Luo, P, Gaspar, F.J, & Oosterlee, C.W. (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