Numerical Analysis: Adaptive methods for subsurface flow
This project focuses on adaptive finite element approximations and a posteriori error estimates for PDE with random data. One would like to model subsurface fluid flow and transport to understand, test, and verify predictions for a variety of applications, including: the propagation of contaminants and pollutants in groundwater, carbon sequestration in deep saline aquifers, and flow in composite biological materials. Such flows are modelled by an elliptic diffusion equation with appropriate boundary conditions where the conductivity, and possibly the force term, are assumed to be random fields on an underlying probability space. In particular, for the propagation of contaminants in groundwater (on the scale of 100s of meters), deterministic predictions of pollutant concentration distributions are impossible given the amount of initial data required. Instead one considers a model where the law of the hydraulic conductivity is assumed to be log-normal due to uncertainty in the geological information (for example, see ). Under these constraints, major obstacles to overcome when performing numerical approximations include the low spatial regularity of the coefficients and the lack of stochastically uniform coercivity and boundedness.
 G. Dagan, “Stochastic modeling of flow and transport: the broad perspective”, in Subsurface Flow and Transport: A Stochastic Approach, Cambridge Univ. Press, Cambridge, 2005.