Numerical Methods for Elliptic Partial Differential Equations and Optimal Control Problems

This thesis studies numerical methods for elliptic partial differential equations and optimal control problems. Second order and fourth order elliptic equations arise in various applications, for example, image processing, thin plates and the Stokes problem in stream function vorticity formulation. The first part focuses on fourth order elliptic equations and a unified convergence analysis framework known as the Hessian discretisation method. The second part focuses on the numerical analysis for optimal control problems governed by second order and fourth order elliptic equations. The second order problems are first considered for which the corresponding framework is called the gradient discretisation method.