#### Degree Name

Master of Science

#### Department or Program

Mathematics

#### Advisor

Budzban, Gregory

#### Abstract

This paper discusses the mathematical theory of finite elements. Using the concepts of inner product spaces, the mathematics of finite element analysis is explained in the context of function spaces. The finite elements as well as the finite element space will be rigorously defined. Furthermore, examples from research in the field of engineering will be explained and viewed through a mathematical lens. This research paper seeks to firmly bridge the applications with mathematical content for the research scientist who generally focuses on the applications. The paper will begin with a review of some basic building blocks for the finite element space followed by validation of the process and construction of finite elements. The use of these elements will be compared. A trial will be done on the computer using a simple differential equation where the use of finite elements will be investigated for accuracy to validate the roof functions used in finite element analysis. Finally a short review of literature illustrating the technique will be presented.