An Introduction to the Theory of Finite Elements

Abstract: A wide variety of partial differential equations that are of interest in industrial applications cannot be solved in closed form. Even those equations for which we do have a well-understood theory may not be solved over domains that aren’t nice enough. The finite element method is a powerful algorithm that provides solutions to both these issues. This talk will provide a very high-level overview of the theory FEM. How do we approximate solutions that we do not know a priori? How can we estimate our error? How are geometric concerns handled by the algorithm? All these considerations will explain why FEM have become so ubiquitous in the world of numerical PDE.