ABSTRACT
In this chapter, we present an abstracted version of the widely used FEM in its standard form, because it is frequently used in this book and is the base for our S-FEM methods to be presented in later chapters. We will focus on some of the essential mathematical and numerical aspects and properties of FEM, but the mathematics language is kept as simple as possible with the objective of helping readers become familiar with the necessary terminologies, mathematics tools, and numerical treatments used in FEM and S-FEM. This also serves the purpose of easy reference in the later chapters of this book, when discussing the properties of our S-FEM models. Operational issues on the general procedure of FEM, discretization of the problem domain, shape function construction, weak-form statement, variational formulation, numerical integration, and formulation of the linear system of equations will be outlined. Theoretical issues on functional spaces, solution existence, uniqueness, error, convergence rate, and major properties of the FEM will also be presented in a concise and easy-tounderstand fashion, but without details on the proofs. More complete and precise discussions on FEM and the detailed modeling techniques can be found in many dedicated books in the open literature, some of which are listed in, for example, Refs. [1-5].
