The Finite Element Method for Boundary Value Problems : Mathematics and Computations

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. 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. 3.3.3.1 Linear differential operators -- 3.3.3.2 Non-linear differential operators -- 3.3.4 The least-squares method -- 3.3.4.1 Self-adjoint and non-self-adjoint linear differential operators -- 3.3.4.2 Non-linear differential operators -- 3.3.5 Collocation method -- 3.4 Approximation Spaces for Various Methods of Approximation -- 3.5 Integral Formulations of BVPs using the Classical Methods of Approximations -- 3.5.1 Self-adjoint differential operators -- 3.5.2 Non-self-adjoint Differential Operators -- 3.5.3 Non-linear Differential Operators 1.2.8 k-version of the finite element method and hpk framework -- 1.3 Summary -- 2: Concepts from Functional Analysis -- 2.1 General Comments -- 2.2 Sets, Spaces, Functions, Functions Spaces, and Operators -- 2.2.1 Hilbert spaces Hk() -- 2.2.2 Definition of scalar product in Hk() space -- 2.2.3 Properties of scalar product -- 2.2.4 Norm of u in Hilbert space Hk() -- 2.2.5 Seminorm of u in Hilbert space Hk() -- 2.2.6 Function spaces -- 2.2.7 Operators -- 2.2.8 Types of operators -- 2.2.9 Energy product -- 2.2.10 Integration by parts (IBP) Content: Introduction Basic Elements from Applied Mathematics Classical Methods of Approximation The Finite Element Method Finite Element Method for Self Adjoint Operators The Finite Element Method for Non-Self Adjoint Operators Finite Element Method for Non-linear operators Basic Elements of Mapping and Interpolation Theory Finite element processes in linear solid and structural mechanics Finite element formulations using principle of virtual work Appendix 3.2 Basic Steps in Classical Methods of Approximation based on Integral Forms -- 3.3 Integral forms using the Fundamental Lemma of the Calculus of Variations -- 3.3.1 The Galerkin method -- 3.3.1.1 Self-adjoint and non-self-adjoint linear differential operators -- 3.3.1.2 Non-linear differential operators -- 3.3.2 The Petrov-Galerkin and weighted-residual methods -- 3.3.2.1 Self-adjoint and non-self-adjoint linear differential operators -- 3.3.2.2 Non-linear differential operators -- 3.3.3 The Galerkin method with weak form 2.3 Elements of Calculus of Variations -- 2.3.1 Concept of the variation of a functional -- 2.3.2 Euler's equation: Simplest variational problem -- 2.3.3 Variation of a functional: some practical aspects -- 2.3.4 Riemann and Lebesgue integrals -- 2.4 Examples of Differential Operators and their Properties -- 2.4.1 Self-adjoint differential operators -- 2.4.2 Non-self-adjoint differential operators -- 2.4.3 Non-linear differential operators -- 2.5 Summary -- 3: Classical Methods of Approximation -- 3.1 Introduction Cover -- Half Title -- Title Page -- Copyright Page -- Dedication -- Table of Contents -- Preface -- About the Authors -- 1: Introduction -- 1.1 General Comments and Basic Philosophy -- 1.2 Basic Concepts of the Finite Element Method -- 1.2.1 Discretization -- 1.2.2 Local approximation -- 1.2.3 Integral forms and algebraic equations over an element -- 1.2.4 Assembly of element equations -- 1.2.5 Computation of the solution -- 1.2.6 Post-processing -- 1.2.7 Remarks