A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system
Prashant Kumar (TU Delft - Aerospace Engineering, Centrum Wiskunde & Informatica (CWI))
Peiyao Luo (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Francisco J. Gaspar (Centrum Wiskunde & Informatica (CWI))
Cornelis W. Oosterlee (TU Delft - Electrical Engineering, Mathematics and Computer Science, Centrum Wiskunde & Informatica (CWI))
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Abstract
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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