Numerical Methods for Darcy Flow Problems with Rough and Uncertain Data

We address two computational challenges for numerical simulations of Darcy flow problems: rough and uncertain data. The rapidly varying and possibly high contrast permeability coefficient for the pressure equation in Darcy flow problems generally leads to irregular solutions, which in turn make standard solution techniques perform poorly. We study methods for numerical homogenization based on localized computations. Regarding the challenge of uncertain data, we consider the problem of forward propagation of uncertainty through a numerical model. More specifically, we consider methods for estimating the failure probability, or a point estimate of the cumulative distribution function (cdf) of a scalar output from the model. The issue of rough coefficients is discussed in Papers I–III by analyzing three aspects of the localized orthogonal decomposition (LOD) method. In Paper I, we define an interpolation operator that makes the localization error independent of the contrast of the coefficient. The conditions for its applicability are studied. In Paper II, we consider time-dependent coefficients and derive computable error indicators that are used to adaptively update the multiscale space. In Paper III, we derive a priori error bounds for the LOD method based on the Raviart–Thomas finite element. The topic of uncertain data is discussed in Papers IV–VI. The main contribution is the selective refinement algorithm, proposed in Paper IV for estimating quantiles, and further developed in Paper V for point evaluation of the cdf. Selective refinement makes use of a hierarchy of numerical approximations of the model and exploits computable error bounds for the random model output to reduce the cost complexity. It is applied in combination with Monte Carlo and multilevel Monte Carlo methods to reduce the overall cost. In Paper VI we quantify the gains from applying selective refinement to a two-phase Darcy flow problem