Doctoral Dissertation Defense: Kevin Wiliamson

We consider three problems related to the numerical solution of the Stokes-Brinkman equations, which model flow through porous media. We first study and apply residual-based a posteriori error estimates to several 2D and 3D problems and use them to drive an adaptive mesh-refinement process.  We demonstrate that the error estimates succeed in identifying regions of the physical domain for which the error is high and for which refinement of the underlying mesh reduces the error without significantly increasing the dimension of the discrete problem.

We then study a stochastic Stokes-Brinkman problem with uncertain permeabilities. We employ a truncated anchored ANOVA decomposition alongside stochastic collocation to estimate the moments of the velocity and pressure solutions. Through an adaptive procedure selecting only the most important ANOVA terms, we reduce the number of collocation points needed for accurate estimation of the statistical moments. However, for even modest stochastic dimensions, the number of collocation points remains too large to perform high-fidelity solves at each point. We use reduced basis methods to alleviate the computational burden by approximating the expensive high-fidelity solves with inexpensive approximate solutions on a low-dimensional space. We furthermore develop and analyze rigorous a posteriori error estimates for the reduced basis approximation. We apply these methods to 2D problems considering both isotropic and anisotropic permeabilities.

In the final problem, we consider an optimal control problem constrained by the Stokes-Brinkman equations. The inverse permeability tensor is treated as the control and we seek to identify a control which minimizes an objective function of tracking type. We derive the Karush-Kuhn-Tucker optimality conditions and show their equivalence to a particular system of nonsmooth nonlinear equations. We solve a few such problems using semismooth Newton methods and present numerical results.

We conclude with a study of block preconditioners for the Stokes-Brinkman system.