Finite element methods for Monge–Ampère type equations

This thesis focuses on the numerical analysis of partial differential equations (PDEs), the main goal being the development and analysis of finite element methods (FEMs) for fully nonlinear elliptic PDEs, particularly Monge-Ampère (MA) and Hamilton-Jacobi-Bellman (HJB) equations. There are two clear distinctions in the approaches that are undertaken in this thesis: firstly, for the approximation of solutions to the MA problem, we implement and analyse a continuous Galerkin (CG) FEM; secondly, to numerically solve the HJB equation, we employ a discontinuous Galerkin (DG) FEM. Though the chosen approaches (CG vs. DG) applied to the MA and HJB type equations are distinct, the equations themselves are related. A longstanding result, proven by N. Krylov in 1987, allows one to characterise the MA equation as a HJB equation. Another important theme of this thesis, motivated by domain assumptions, necessary for the well-posedness of MA type problems, and oblique boundary-value problems is the implementation and analysis of FEMs on domains with curved boundaries. In the case of DG methods, where the consistency of the method plays a key role in obtaining a priori error estimates for the numerical solution, this quantitative consideration requires new techniques to extend the existing DG framework. he main contributions of this thesis are new results concerning the existence and uniqueness of numerical solutions to CG and DG finite element methods on curved domains, for both fully nonlinear elliptic equations, and linear elliptic equations in nondivergence form, with Dirichlet and oblique derivative boundary conditions, as well as optimal a priori error estimates. Furthermore, we prove several key results from the theory of finite elements in the context of curved finite elements, that do not appear to be available in the literature.