Оглавление
Mohammad Asadzadeh. An Introduction to the Finite Element Method for Differential Equations
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
An Introduction to the Finite Element Method for Differential Equations
Preface
Acknowledgments
1 Introduction
1.1 Preliminaries
Example 26.1
1.2 Trinities for Second‐Order PDEs
1.3 PDEs in, Further Classifications
Definition 1.1
Example 1.10
Definition 1.3
1.4 Differential Operators, Superposition
Definition 1.4
1.4.1 Exercises
1.5 Some Equations of Mathematical Physics
1.5.1 The Poisson Equation
1.5.2 The Heat Equation. 1.5.2.1 A Model Problem for the Stationary Heat Equation in
Remark 1.3
1.5.2.2 Fourier's Law of Heat Conduction, Derivation of the Heat Equation
Lemma 1.1
Proof:
1.5.3 The Wave Equation
1.5.3.1 The Vibrating String, Derivation of the Wave Equation in
1.5.4 Exercises
2 Mathematical Tools
2.1 Vector Spaces. Definition 2.1
Definition 2.2
Definition 2.3 (Scalar product)
Definition 2.4
Example 2.1
Definition 2.5 (Orthogonality)
Definition 2.6
Definition 2.7 (The ‐space and the ‐norm)
Example 2.2
Proof:
Remark 2.1
Example 2.3
Proof:
Remark 2.2
2.1.1 Linear Independence, Basis, and Dimension
Definition 2.8
Example 2.4
Definition 2.9
Example 2.5
Remark 2.3
Example 2.6
Example 2.7
Example 2.8
Definition 2.10
Definition 2.11
Example 2.9
Lemma 2.1
Theorem 2.1
Theorem 2.2
Theorem 2.3
Theorem 2.4
2.2 Function Spaces. 2.2.1 Spaces of Differentiable Functions
Definition 2.12
Example 2.10
Definition 2.13
Example 2.11
2.2.2 Spaces of Integrable Functions
Remark 2.4
2.2.3 Weak Derivative
Definition 2.14
Example 2.12
2.2.4 Sobolev Spaces. Definition 2.15
Remark 2.5
2.2.5 Hilbert Spaces
Definition 2.16
Theorem 2.5 (Sobolev embedding theorem)
2.3 Some Basic Inequalities
Definition 2.17
Theorem 2.6 (Minkowski's and Hölder's inequalities)
Theorem 2.7 (Poincaré inequality)
Proof:
Remark 2.6
Theorem 2.8 (Trace theorem)
Lemma 2.2 (Grönwall's lemma)
Proof:
Theorem 2.9 (Sobolev inequality)
2.4 Fundamental Solution of PDEs1
Remark 2.7
Theorem 2.10
2.4.1 Green's Functions
Definition 2.18
Theorem 2.11 (Green's function for Laplace's equation)
2.5 The Weak/Variational Formulation
Lemma 2.3 (Green's formula)
Remark 2.8
Definition 2.19
Definition 2.20
2.6 A Framework for Analytic Solution in 1d
2.6.1 The Variational Formulation in
Theorem 2.12
Proof:
Corollary 2.1
2.6.2 The Minimization Problem in
Theorem 2.13
Proof:
Corollary 2.2
2.6.3 A Mixed Boundary Value Problem in
Theorem 2.14
Proof:
Remark 2.9
2.7 An Abstract Framework. Example 2.13
Definition 2.21
Definition 2.22
Example 2.14
Definition 2.23
Definition 2.24
Theorem 2.15
Definition 2.25
Theorem 2.16
Definition 2.26
Example 2.15
2.7.1 Riesz and Lax–Milgram Theorems
Theorem 2.17
Proof:
Proposition 2.1
Theorem 2.18 (Riesz representation theorem)
Theorem 2.19 (Lax–Milgram theorem)
Remark 2.10
Remark 2.11
Example 2.16
Example 2.17
2.8 Exercises
Note
3 Polynomial Approximation/Interpolation in 1d
3.1 Finite Dimensional Space of Functions on an Interval
3.2 An Ordinary Differential Equation (ODE)
3.2.1 Forward Euler Method to Solve IVP
3.2.2 Variational Formulation for IVP
Definition 3.1
3.2.3 Galerkin Method for IVP
3.3 A Galerkin Method for (BVP)
Remark 3.2
Remark 3.3
3.3.1 An Equivalent Finite Difference Approach
Remark 3.4
Definition 3.2
Proposition 3.1
Proof:
Example 3.2
3.4 Exercises
3.5 Polynomial Interpolation in 1d
Definition 3.3
Example 3.3
Example 3.4 (Linear interpolation on an interval)
Remark 3.5
Question 3.1
Theorem 3.1
Proof:
Remark 3.6
Theorem 3.2
Proof:
Lemma 3.1
Proof:
Definition 3.4
Definition 3.5
Remark 3.7
Theorem 3.3
Proof:
3.5.1 Lagrange Interpolation
Definition 3.6 (Cardinal functions)
Example 3.5
Example 3.6
Example 3.7
Definition 3.7 (Taylor's theorem)
Theorem 3.4
Proof:
Theorem 3.5 (Generalized Rolle's theorem)
3.6 Orthogonal‐ and L2‐Projection
3.6.1 The ‐Projection onto the Space of Polynomials
Definition 3.8
Lemma 3.2
Proof:
3.7 Numerical Integration, Quadrature Rule
3.7.1 Composite Rules for Uniform Partitions
Remark 3.8
Remark 3.9
3.7.2 Gauss Quadrature Rule
Example 3.8
Example 3.9
Example 3.10
Theorem 3.6
Proof:
3.8 Exercises
4 Linear Systems of Equations
4.1 Direct Methods
Remark 4.1
Example 4.1
Remark 4.2
Definition 4.1
Example 4.2
Proposition 4.1
4.1.1 LU Factorization of an Matrix
Example 4.3
Remark 4.3
Theorem 4.1 Cholesky's method
4.2 Iterative Methods
Definition 4.2
4.2.1 Jacobi Iteration
4.2.2 Convergence Criterion
Example 4.4
4.2.3 Gauss–Seidel Iteration
Example 4.5
4.2.4 The Successive Over‐Relaxation Method (S.O.R.)
Remark 4.4
4.2.5 Abstraction of Iterative Methods
4.2.5.1 Questions
4.2.6 Jacobi's Method
Example 4.6
Example 4.7
4.2.7 Gauss–Seidel's Method
4.2.7.1 Relaxation
4.3 Exercises
5 Two‐Point Boundary Value Problems
5.1 The Finite Element Method (FEM)
5.2 Error Estimates in the Energy Norm
Theorem 5.1
Proof:
Theorem 5.2
Proof:
Remark 5.1
Remark 5.2
Theorem 5.3 An a posteriori error estimate
Proof:
Remark 5.3
5.2.1 Adaptivity
5.3 FEM for Convection–Diffusion–Absorption BVPs
Example 5.1
Remark 5.4
Example 5.2
Remark 5.5
Remark 5.6
Example 5.3
5.4 Exercises
Note
6 Scalar Initial Value Problems
6.1 Solution Formula and Stability. Theorem 6.1
Proof:
Theorem 6.2 (Stability estimates)
Proof:
Remark 6.1
6.2 Finite Difference Methods for IVP
Example 6.1
6.3 Galerkin Finite Element Methods for IVP
6.3.1 The Continuous Galerkin Method
Example 6.2
Example 6.3
6.3.1.1 The cG(1) Algorithm
6.3.1.2 The cG() Method
6.3.2 The Discontinuous Galerkin Method
Example 6.4 dG(0)
Remark 6.2
6.4 A Posteriori Error Estimates
6.4.1 A Posteriori Error Estimate for cG(1)
6.4.1.1 The Dual Problem
Theorem 6.3 A posteriori error estimate for cG(1)
Proof:
Theorem 6.4 Convergence order
Proof:
Remark 6.3
6.4.2 A Posteriori Error Estimate for dG(0)
Theorem 6.5
Proof:
6.4.3 Adaptivity for dG(0)
6.4.3.1 An Adaptivity Algorithm
Remark 6.4
6.5 A Priori Error Analysis
6.5.1 A Priori Error Estimates for the dG(0) Method
Lemma 6.1
Proof:
Lemma 6.2
Theorem 6.6
Proof:
Remark 6.5
Proof of Lemma 6.2
6.6 The Parabolic Case (a(t) ≥ 0)
Theorem 6.7
Main steps in the proof
Lemma 6.3
Proof:
Theorem 6.8 The modified a posteriori estimate for dG(0)
6.6.1 An Example of Error Estimate
Example 6.5
Proof:
6.7 Exercises
7 Initial Boundary Value Problems in 1d
7.1 The Heat Equation in 1d
Example 7.1
Remark 7.1
7.1.1 Stability Estimates
Theorem 7.1
Proof:
Theorem 7.2 (Stability of the homogeneous heat equation)
Remark 7.2
Proof:
Remark 7.3
Theorem 7.3 (An energy estimate)
Proof:
Remark 7.4
7.1.2 FEM for the Heat Equation
7.1.3 Error Analysis
Theorem 7.4 (cG(1)cG(1) a posteriori error estimate)
Proof:
Example 7.2 (Error estimates in a simple case for cG(2))
Example 7.3 (The equation of an elastic beam)
7.1.4 Exercises
7.2 The Wave Equation in 1d
Theorem 7.5
Proof:
7.2.1 Wave Equation as a System of Hyperbolic PDEs
7.2.2 The Finite Element Discretization Procedure
7.2.3 Exercises
7.3 Convection–Diffusion Problems
Example 7.4
Remark 7.5
Example 7.5 (The boundary layer)
7.3.1 Finite Element Method
Remark 7.6
7.3.2 The Streamline‐Diffusion Method (SDM)
Remark 7.7
7.3.3 Exercises
8 Approximation in Several Dimensions. 8.1 Introduction
Lemma 8.1 (Green's formula)
Proof:
8.2 Piecewise Linear Approximation in 2d
8.2.1 Basis Functions for the Piecewise Linears in
Definition 8.1
Definition 8.2
Example 8.1
Example 8.2
8.3 Constructing Finite Element Spaces
Definition 8.3
Definition 8.4
Example 8.3
Remark 8.1
8.4 The Interpolant
Definition 8.5
Example 8.4
Lemma 8.2 (The properties of the local interpolation)
Proof:
Definition 8.6
Definition 8.7
Definition 8.8
Example 8.5
8.4.1 Error Estimates for Piecewise Linear Interpolation
Definition 8.9
Proposition 8.1 (Polynomial approximation)
Lemma 8.3 (Bramble–Hilbert lemma)
Proof:
Corollary 8.1
Theorem 8.1
Proof:
Definition 8.10 (Shape‐regularity)
Theorem 8.2 (Local interpolation)
Proof:
Theorem 8.3 (Global interpolation)
Proof:
Theorem 8.4 (Interpolation in )
Proof:
Theorem 8.5
Remark 8.2
8.5 The L2 (Revisited) and Ritz Projections
Definition 8.11
Theorem 8.6
Proof:
Theorem 8.7
Proof:
8.5.1 The Ritz or Elliptic Projection
Definition 8.12
8.6 Exercises
9 The Boundary Value Problems in N
9.1 The Poisson Equation
9.1.1 Weak Stability
9.1.2 Error Estimates for the CG(1) FEM
Theorem 9.1 (cG(1) a priori error estimate for the gradient )
Proof:
Lemma 9.1 (Regularity lemma)
Theorem 9.2 (cG(1) a priori error estimate for the solution )
Proof:
Corollary 9.1 (A priori strong stability estimate)
Theorem 9.3 (cG(1) a posteriori error estimate)
Proof:
Corollary 9.2 (A posteriori strong stability estimate)
9.1.3 Proof of the Regularity Lemma. Proof:
9.2 Stationary Convection–Diffusion Equation
9.2.1 The Elliptic Case
Definition 9.1
9.2.1.1 A Brief Note on Distributions. Definition 9.2
Theorem 9.4 ([76] or [120])
Definition 9.3
Assumptions 9.1
Definition 9.4 (Smoothing property)
Remark 9.1
9.2.2 Error Estimates
Theorem 9.5 (The gradient/‐ estimate)
Proof:
Theorem 9.6 (The error estimate)
Proof:
Theorem 9.7
Proof:
9.3 Hyperbolicity Features
9.3.1 Convection Dominating Case
9.3.2 The SD Method for Convection Diffusion Problem
Proposition 9.6
Proof:
9.3.3 Stability Estimates. Theorem 9.8
Proof:
Remark 9.2
9.3.4 Error Estimates for Convention Dominating in
Proposition 9.7
Proof:
Theorem 9.9
Proof:
Theorem 9.10
Proof:
Remark 9.3
9.4 Exercises
10 The Initial Boundary Value Problems in N
10.1 The Heat Equation in N
10.1.1 The Fundamental Solution
10.1.2 Stability
Theorem 10.1 (Energy estimates)
Proof
10.1.3 The Finite Element for Heat Equation. 10.1.3.1 The Semidiscrete Problem
Theorem 10.2
Proof
Theorem 10.3
Proof
10.1.4 A Fully Discrete Algorithm
10.1.5 The Discrete Equations
10.1.6 A Priori Error Estimate: Fully Discrete Problem
Theorem 10.4
10.2 The Wave Equation in d
Theorem 10.5 (Conservation of energy)
Proof
10.2.1 The Weak Formulation
10.2.2 The Semidiscrete Problem
10.2.2.1 A Priori Error Estimates for the Semidiscrete Problem. Theorem 10.6
Proof
Theorem 10.7
Proof
10.2.3 The Fully Discrete Problem
10.2.3.1 Finite Elements for the Fully Discrete Problem
Remark 10.1
10.2.4 Error Estimate for cG(1) Theorem 10.8 (An a priori error estimate)
Proof
10.3 Exercises
Appendix A Answers to Some Exercises. Chapter 1. Exercise Section 1.4.1
Chapter 1. Exercise Section 1.5.4
Chapter 2. Exercise Section 2.11
Chapter 3. Exercise Section 3.5
Chapter 3. Exercise Section 3.8
Chapter 4. Exercise Section 4.3
Chapter 5. Exercise Section 5.4
Chapter 6. Exercise Section 6.7
Chapter 7. Exercise Section 7.2.3
Chapter 7. Exercise Section 7.3.3
Chapter 9. Poisson Equation. Exercise Section 9.4
Chapter 10. IBVPs: Exercise Section 10.3
Appendind B Algorithms and Matlab Codes
B.1 A Matlab Code to Compute the Mass Matrix M for a Nonuniform Mesh
B.1.1 A Matlab Routine to Compute the Load Vector b
B.2 Matlab Routine to Compute the L2‐Projection
B.2.1 A Matlab Routine for the Composite Midpoint Rule
B.2.2 A Matlab Routine for the Composite Trapezoidal Rule
B.2.3 A Matlab Routine for the Composite Simpson's Rule
B.3 A Matlab Routine Assembling the Stiffness Matrix
B.4 A Matlab Routine to Assemble the Convection Matrix
B.5 Matlab Routine for Forward‐, Backward‐Euler, and Crank–Nicolson
B.6 A Matlab Routine for Mass‐Matrix in 2d
B.7 A Matlab Routine for a Poisson Assembler in 2d
Appendix C Sample Assignments. C.1 Assignment 1
C.2 Assignment 2. C.2.1 Grading Policy of the Assignment
C.2.2 Theory
C.2.3 Selected Applications
C.2.3.1 Convection–Diffusion–Absorption/Reaction
C.2.3.2 Electrostatics
C.2.3.3 Fluid Flow
C.2.3.4 Heat Conduction
C.2.3.5 Quantum Physics
Appendix D Symbols. D.1 Table of Symbols
Bibliography
Index
WILEY END USER LICENSE AGREEMENT
