Introduction to the Finite Element Method for Differential Equations

Master the finite element method with this masterful and practical volume
 
An Introduction to the Finite Element Method (FEM) for Differential Equations provides readers with a practical and approachable examination of the use of the finite element method in mathematics. Author Mohammad Asadzadeh covers basic FEM theory, both in one-dimensional and higher dimensional cases.
 
The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, An Introduction to the Finite Element Method covers topics including:
* An introduction to basic ordinary and partial differential equations
* The concept of fundamental solutions using Green's function approaches
* Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations
* Higher-dimensional interpolation procedures
* Stability and convergence analysis of FEM for differential equations
 
This book is ideal for upper-level undergraduate and graduate students in natural science and engineering. It belongs on the shelf of anyone seeking to improve their understanding of differential equations.

List of contents

Preface xi
 
Acknowledgments xiii
 
1 Introduction 1
 
1.1 Preliminaries 2
 
1.2 Trinities for Second-Order PDEs 4
 
1.3 PDEs in R^n, Further Classifications 10
 
1.4 Differential Operators, Superposition 12
 
1.4.1 Exercises 14
 
1.5 Some Equations of Mathematical Physics 15
 
1.5.1 The Poisson Equation 16
 
1.5.2 The Heat Equation 17
 
1.5.2.1 A Model Problem for the Stationary Heat Equation in 1d 17
 
1.5.2.2 Fourier's Law of Heat Conduction, Derivation of the Heat Equation 18
 
1.5.3 The Wave Equation 21
 
1.5.3.1 The Vibrating String, Derivation of the Wave Equation in 1d 21
 
1.5.4 Exercises 24
 
2 Mathematical Tools 27
 
2.1 Vector Spaces 27
 
2.1.1 Linear Independence, Basis, and Dimension 30
 
2.2 Function Spaces 33
 
2.2.1 Spaces of Differentiable Functions 33
 
2.2.2 Spaces of Integrable Functions 34
 
2.2.3 Weak Derivative 35
 
2.2.4 Sobolev Spaces 36
 
2.2.5 Hilbert Spaces 37
 
2.3 Some Basic Inequalities 38
 
2.4 Fundamental Solution of PDEs 41
 
2.4.1 Green's Functions 43
 
2.5 The Weak/Variational Formulation 44
 
2.6 A Framework for Analytic Solution in 1d 46
 
2.6.1 The Variational Formulation in 1d 48
 
2.6.2 The Minimization Problem in 1d 51
 
2.6.3 A Mixed Boundary Value Problem in 1d 52
 
2.7 An Abstract Framework 54
 
2.7.1 Riesz and Lax-Milgram Theorems 57
 
2.8 Exercises 63
 
3 Polynomial Approximation/Interpolation in 1d 67
 
3.1 Finite Dimensional Space of Functions on an Interval 67
 
3.2 An Ordinary Differential Equation (ODE) 71
 
3.2.1 Forward Euler Method to Solve IVP 71
 
3.2.2 Variational Formulation for IVP 72
 
3.2.3 Galerkin Method for IVP 73
 
3.3 A Galerkin Method for (BVP) 74
 
3.3.1 An Equivalent Finite Difference Approach 79
 
3.4 Exercises 82
 
3.5 Polynomial Interpolation in 1d 83
 
3.5.1 Lagrange Interpolation 90
 
3.6 Orthogonal- and L2-Projection 94
 
3.6.1 The L2-Projection onto the Space of Polynomials 94
 
3.7 Numerical Integration, Quadrature Rule 96
 
3.7.1 Composite Rules for Uniform Partitions 98
 
3.7.2 Gauss Quadrature Rule 101
 
3.8 Exercises 105
 
4 Linear Systems of Equations 109
 
4.1 Direct Methods 110
 
4.1.1 LU Factorization of an n × n Matrix A 113
 
4.2 Iterative Methods 115
 
4.2.1 Jacobi Iteration 115
 
4.2.2 Convergence Criterion 116
 
4.2.3 Gauss-Seidel Iteration 117
 
4.2.4 The Successive Over-Relaxation Method (S.O.R.) 119
 
4.2.5 Abstraction of Iterative Methods 120
 
4.2.5.1 Questions 120
 
4.2.6 Jacobi's Method 120
 
4.2.7 Gauss-Seidel's Method 121
 
4.2.7.1 Relaxation 121
 
4.3 Exercises 122
 
5 Two-Point Boundary Value Problems 125
 
5.1 The Finite Element Method (FEM) 125
 
5.2 Error Estimates in the Energy Norm 127
 
5.2.1 Adaptivity 132
 
5.3 FEM for Convection-Diffusion-Absorption BVPs 132
 
5.4 Exercises 140
 
6 Scalar Initial Value Problems 147
 
6.1 Solution Formula and Stability 147
 
6.2 Finite Difference Methods for IVP 149
 
6.3 Galerkin Finite Element Methods for IVP 151
 
6.3.1 The Continuous Galerkin Method 152
 
6.3.1.1 The cG(1) Algorithm 154
 
6.3.1.2 The cG(q) Method 154
 
6.3.2 The Discontinuous Galerkin Method 155
 
6.4 A Posteriori Error Estimates 156
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About the author

MOHAMMAD ASADZADEH, PHD is Professor of Applied Mathematics at the Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg. His primary research interests include the numerical analysis of hyperbolic pdes, as well as convection-diffusion and integro-differential equations.

Summary

Master the finite element method with this masterful and practical volume

An Introduction to the Finite Element Method (FEM) for Differential Equations provides readers with a practical and approachable examination of the use of the finite element method in mathematics. Author Mohammad Asadzadeh covers basic FEM theory, both in one-dimensional and higher dimensional cases.

The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, An Introduction to the Finite Element Method covers topics including:
* An introduction to basic ordinary and partial differential equations
* The concept of fundamental solutions using Green's function approaches
* Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations
* Higher-dimensional interpolation procedures
* Stability and convergence analysis of FEM for differential equations

This book is ideal for upper-level undergraduate and graduate students in natural science and engineering. It belongs on the shelf of anyone seeking to improve their understanding of differential equations.

Product details

• Authors: M Asadzadeh, Mohammad Asadzadeh
• Publisher: Wiley, John and Sons Ltd
• Languages: English
• Product format: Hardback
• Released: 31.08.2020
• EAN: 9781119671640
• ISBN: 978-1-119-67164-0
• No. of pages: 352
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Mathematik, MATHEMATICS / Differential Equations / General, Mathematics, Biochemische Verfahrenstechnik, Differentialgleichungen, Chemical Engineering, chemische Verfahrenstechnik, numerische Methoden, Differential equations, Biochemical Engineering, Numerical Methods, Finite-Element-Methode