OF THE DISSERTATION Geometric Aspects of Discretized Classical Field Theories: Extensions to Finite Element Exterior Calculus, Noether Theorems, and the Geodesic Finite Element Method by Joe Salamon Doctor of Philosophy in Physics University of California San Diego, 2016 Professor Melvin Leok, Chair Professor Michael Holst, Co-Chair In this dissertation, I will discuss and explore the various theoretical pillars required to investigate the world of discretized gauge theories in a purely classical setting, with the long-term aim of achieving a fully-fledged discretization of General Relativity (GR). I will present some results on the geometric framework of finite element exterior calculus (FEEC); in particular, I will elaborate on integrating metric structures within the framework and categorize the dual spaces of the various spaces of polynomial differential forms PrΛ(R). I will also introduce the Rapetti construction, and then demonstrate the general issues with providing geometric interpretations to polynomial order within FEEC. After a brief pedagogical detour through Noether’s theorems, I will apply all of the above into discretizations of electromagnetism and linearized GR. I will conclude with an excursion into the geodesic finite element method (GFEM) as a way to work with nonlinear manifolds.