AbstractsMathematics

PDE-constrained optimization is an important area in the field of numerical analysis, with problems arising in a wide variety of applications including optimal design, optimal control and parameter estimation. The aim of such problems is to minimize a functional J(u,d) whilst adhering to constraints posed by a system of partial differential equations (PDE), with u and d used respectively to denote the state and control of the system. In this thesis, we describe the steady-state generalized Stokes equations for incompressible fluids. We proceed to derive the weak formulation of the problem, and show that the resulting system may be written in terms of a mixed formulation of the Stokes problem. Based on this formulation, the problem is discretized through use of the Galerkin finite element method, before investigating control problems based on the generalized Stokes equations, along with numerical experimentation. This work will be used to achieve the main aim of this thesis, namely the exploration and investigation of solution methods for optimal control problems constrained by non- Newtonian flow. Ultimately, an iterative solution method designed for such problems coupled with an appropriate preconditioning strategy will be described and analyzed, and used to produce effective numerical results.