Numerical methods for fully nonlinear second order partial differential equations

This dissertation concerns the numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). The numerical methods and analysis are based on a new concept of weak solutions called moment solutions, which unlike viscosity solutions, are defined by a constructive method called the vanishing moment method. The main idea of the vanishing moment method is to approximate fully nonlinear second order PDEs by a family of fourth order quasi-linear PDEs. Because the method is constructive, we can develop a wealth of convergent numerical discretization methods to approximate fully nonlinear second order PDEs. We first study the numerical approximation of the prototypical second order fully nonlilnear PDE, the Monge-Ampère equation, det(D²u) = f (> 0), using C¹ finite element methods, spectral Galerkin methods, mixed finite element methods, and a nonconforming Morley finite element method. We then generalize the analysis to other fully nonlinear second order PDEs including the nonlinear balance equation, a nonlinear formulation of semigeostrophic flow equations, and the equation of prescribed Gauss curvature.