Written by two well-respected experts in the field, The Finite Element Method for Boundary Value Problems: Mathematics and Computations bridges the gap between applied mathematics and application-oriented computational studies using FEM. Mathematically rigorous, the FEM is presented as a method of approximation for differential operators that are mathematically classified as self-adjoint, non-self-adjoint, and non-linear, thus addressing totality of all BVPs in various areas of engineering, applied mathematics, and physical sciences. These classes of operators are utilized in various methods of approximation: Galerkin method, Petrov-Galerkin Method, weighted residual method, Galerkin method with weak form, least squares method based on residual functional, etc. to establish unconditionally stable finite element computational processes using calculus of variations. Readers are able to grasp the mathematical foundation of finite element method as well as its versatility of applications. h-, p-, and k-versions of finite element method, hierarchical approximations, convergence, error estimation, error computation, and adaptivity are additional significant aspects of this book.

Table of Contents:
1 Introduction General Comments and Basic Philosophy Basic Concepts of the Finite Element Method Summary Concepts from Functional Analysis General Comments Sets, Spaces, Functions, Functions Spaces, and Operators Elements of Calculus of Variations Examples of Differential Operators and their Properties 2.5 Summary Classical Methods of Approximation Introduction Basic Steps in Classical Methods of Approximation based on Integral Forms Integral forms using the Fundamental Lemma of the Calculus of Variations Approximation Spaces for Various Methods of Approximation Integral Formulations of BVPs using the Classical Methods of Approximations Numerical Examples Summary The Finite Element Method Introduction Basic steps in the finite element method Summary Self-Adjoint Differential Operators Introduction One-dimensional BVPs in a single dependent variable 5.3 Two-dimensional boundary value problems 5.4 Three-dimensional boundary value problems 5.5 Summary 6 Non-Self-Adjoint Differential Operators 6.1 Introduction 6.2 1D convection-diffusion equation 6.3 2D convection-diffusion equation 6.4 Summary 7 Non-Linear Differential Operators 7.1 Introduction 7.2 One dimensional Burgers equation 7.3 Fully developed ow of Giesekus Fluid between parallel plates (polymer flow) 7.4 2D steady-state Navier-Stokes equations 7.5 2D compressible Newtonian fluid Flow 7.6 Summary 8 Basic Elements of Mapping and Interpolation Theory 8.1 Mapping in one dimension 8.2 Elements of interpolation theory over 8.4 Local approximation over : quadrilateral elements 8.5 2D p-version local approximations 8.6 2D approximations for quadrilateral elements 8.10 Serendipity family of interpolations 8.11 Interpolation functions for 3D elements 8.12 Summary 9 Linear Elasticity using the Principle of Minimum Total Potential Energy 9.1 Introduction 9.2 New notation 9.3 Approach 9.4 Element equations 9.5 Finite element formulation for 2D linear elasticity 9.6 Summary 10 Linear and Nonlinear Solid Mechanics using the Principle of Virtual Displacements 10.1 Introduction 10.2 Principle of virtual displacements 10.3 Virtual work statements 10.4 Solution method 10.5 Finite element formulation for 2D solid continua 10.6 Finite element formulation for 3D solid continua 10.7 Axisymmetric solid finite elements 10.8 Summary 11 Additional Topics in Linear Structural Mechanics 11.1 Introduction 11.2 1D axial spar or rod element in R1 (1D space) 11.3 1D axial spar or rod element in R2 11.4 1D axial spar or rod element in R3 (3D space) 11.5 The Euler-Bernoulli beam element 11.6 Euler-Bernoulli frame elements in R2 11.7 The Timoshenko beam elements 11.8 Finite element formulations in R2 and R3 11.9 Summary 12 Convergence, Error Estimation, and Adaptivity 12.1 Introduction 12.2 h-, p-, k-versions of FEM and their convergence 12.3 Convergence and convergence rate 12.4 Error estimation and error computation 12.5 A priori error estimation 12.6 Model problems 12.7 A posteriori error estimation and computation 12.8 Adaptive processes in finite element computations 12.9 Summary Appendix A: Numerical Integration using Gauss Quadrature A.1 Gauss quadrature in R1, R2 and R3 A.2 Gauss quadrature over triangular domains