This book is an elementary text on the finite element method. It is aimed at engineering and science undergraduates with no previous knowledge of the method, and deliberately attempts to keep the mathematics of the subject as straightforward as possible. It is assumed that the reader does understand the basic concepts and equations of elasticity and thermal heat flow, and is familiar with simple matrix algebra. The finite element method has developed into a sophisticated and apparently complex technique, but the fundamental principles are simple and easily understood if treated in a direct and logical manner. The procedure may be introduced in two different ways. Firstly, it may be presented as a mathematical tool, with the theory developed as a general procedure for obtaining approximate solutions to elliptic partial differential equations. Alternatively, the method may be introduced less rigorously, aimed sepcifically at those areas where it is most commonly used. This second approach is adopted here. In many cases an engineer will concentrate on either field problems or elasticity problems, and for this reason the two areas are not generally intermixed in this text. Each is dealt with in separate chapters and if not required, the unwanted chapters may be omitted. For this reason, the chapters before and after the specialized ones tend to be general, with advanced applications of the finite element method covered later in the book. Thus, this text attempts to introduce the finite element method in a simple manner, and places particular emphasis on the practical side of the procedure. It does not discuss how the method is programmed into the computer, that usually need not concern the engineer, although those features of the program which effect the efficient running and execution of the analysis are highlighted.