About the book
This book will cover the numerical and theoretical foundation of the finite element method as a general numerical method for solving ordinary and partial differential equations. It will address the Galerkin method for one-, two- and three-dimensional differential equations (including parabolic, hyperbolic and elliptic partial differential equations) and also convergence and stability criterion. Several continuous and discontinuous finite element methods and adaptive refinement strategies will also be introduced as well as the error estimation techniques for typical finite element methods.
Throughout the text the implementation and analysis of the involved algorithms will be emphasized. In addition, some theoretical aspects as existence, uniqueness, stability and convergence will be discussed together with some of the most important applications of finite elements and its basic methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.
One objective of this book is to show the reader the basic tools for analyzing finite element methods while proofs of stability and convergence of the method will be given in more detail. Another objective is to familiarize the reader with the coding issues of finite element methods: data structure, construction of local matrices, and assembling of the global matrix. Several computational examples will also be provided. Finally, by presenting specific applications of finite element methods to important engineering problems, we hope to convince the reader that the finite element method is a competitive approach for solving many scientific problems.