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E-raamat: Introduction to the Finite Element Method for Differential Equations

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  • Ilmumisaeg: 27-Aug-2020
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119671664
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Introduction to the Finite Element Method for Differential EquationsSuurem pilt
  • Formaat: EPUB+DRM
  • Ilmumisaeg: 27-Aug-2020
  • Kirjastus: John Wiley & Sons Inc
  • Keel: eng
  • ISBN-13: 9781119671664
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"The objective of this book is two-fold. The first objective is to construct, as much as possible, stable finite element schemes without affecting accuracy. The second objective is to derive convergence of the numerical schemes up to maximal available regularity of the exact solution. The first two chapters of the book cover existence, uniqueness and stability as well as the working environment, such as vector and function spaces and principle mathematical inequalities. Chapters 3 and 4 cover the approximation procedure with piecewise linears, interpolation, numerical integration and numerical solution of linear system of equations. Chapters 5 through 7 are devoted to the finite element approximations for the one-space dimensional, boundary value problems, initial value problems, and initial-boundary value problems. Finally, Chapters 8 through 10 are an extension of Chapters 3 and 5-7 to higher spatial dimensions. This book is a great resource for upper undergraduates and graduates in applied math, engineering and natural sciences, as well as researchers in industry and academia in need of finite element approximation techniques. Researchers in industry and academia in need of finite element approximation techniques. Some advanced classes in second year."--

The objective of this book is two-fold. The first objective is to construct, as much as possible, stable finite element schemes without affecting accuracy. The second objective is to derive convergence of the numerical schemes up to maximal available regularity of the exact solution. The first two chapters of the book cover existence, uniqueness and stability as well as the working environment, such as vector and function spaces and principle mathematical inequalities. Chapters 3 and 4 cover the approximation procedure with piecewise linears, interpolation, numerical integration and numerical solution of linear system of equations. Chapters 5 through 7 are devoted to the finite element approximations for the one-space dimensional, boundary value problems, initial value problems, and initial-boundary value problems. Finally, Chapters 8 through 10 are an extension of Chapters 3 and 5-7 to higher spatial dimensions. This book is a great resource for upper undergraduates and graduates in applied math, engineering and natural sciences, as well as researchers in industry and academia in need of finite element approximation techniques. 

Researchers in industry and academia in need of finite element approximation techniques. Some
advanced classes in second year.
Preface xi
Acknowledgments xiii
1 Introduction
1(26)
1.1 Preliminaries
2(2)
1.2 Trinities for Second-Order PDEs
4(6)
1.3 PDEs in R", Further Classifications
10(2)
1.4 Differential Operators, Superposition
12(3)
1.4.1 Exercises
14(1)
1.5 Some Equations of Mathematical Physics
15(12)
1.5.1 The Poisson Equation
16(1)
1.5.2 The Heat Equation
17(1)
1.5.2.1 A Model Problem for the Stationary Heat Equation in Id
17(1)
1.5.2.2 Fourier's Law of Heat Conduction, Derivation of the Heat Equation
18(3)
1.5.3 The Wave Equation
21(1)
1.5.3.1 The Vibrating String, Derivation of the Wave Equation in Id
21(3)
1.5.4 Exercises
24(3)
2 Mathematical Tools
27(40)
2.1 Vector Spaces
27(6)
2.1.1 Linear Independence, Basis, and Dimension
30(3)
2.2 Function Spaces
33(5)
2.2.1 Spaces of Differentiable Functions
33(1)
2.2.2 Spaces of Integrable Functions
34(1)
2.2.3 Weak Derivative
35(1)
2.2.4 Sobolev Spaces
36(1)
2.2.5 Hilbert Spaces
37(1)
2.3 Some Basic Inequalities
38(3)
2.4 Fundamental Solution of PDEs
41(3)
2.4.1 Green's Functions
43(1)
2.5 The Weak/Variational Formulation
44(2)
2.6 A Framework for Analytic Solution in Id
46(8)
2.6.1 The Variational Formulation in Id
48(3)
2.6.2 The Minimization Problem in Id
51(1)
2.6.3 A Mixed Boundary Value Problem in Id
52(2)
2.7 An Abstract Framework
54(9)
2.7.1 Riesz and Lax-Milgram Theorems
57(6)
2.8 Exercises
63(4)
3 Polynomial Approximation/Interpolation in Id
67(42)
3.1 Finite Dimensional Space of Functions on an Interval
67(4)
3.2 An Ordinary Differential Equation (ODE)
71(3)
3.2.1 Forward Euler Method to Solve IVP
71(1)
3.2.2 Variational Formulation for IVP
72(1)
3.2.3 Galerkin Method for IVP
73(1)
3.3 A Galerkin Method for (BVP)
74(8)
3.3.1 An Equivalent Finite Difference Approach
79(3)
3.4 Exercises
82(1)
3.5 Polynomial Interpolation in Id
83(11)
3.5.1 Lagrange Interpolation
90(4)
3.6 Orthogonal- and L2-Projection
94(2)
3.6.1 The L2-Projection onto the Space of Polynomials
94(2)
3.7 Numerical Integration, Quadrature Rule
96(9)
3.7.1 Composite Rules for Uniform Partitions
98(3)
3.7.2 Gauss Quadrature Rule
101(4)
3.8 Exercises
105(4)
4 Linear Systems of Equations
109(16)
4.1 Direct Methods
110(5)
4.1.1 LU Factorization of an n × n Matrix A
113(2)
4.2 Iterative Methods
115(7)
4.2.1 Jacobi Iteration
115(1)
4.2.2 Convergence Criterion
116(1)
4.2.3 Gauss-Seidel Iteration
117(2)
4.2.4 The Successive Over-Relaxation Method (S.O.R.)
119(1)
4.2.5 Abstraction of Iterative Methods
120(1)
4.2.5.1 Questions
120(1)
4.2.6 Jacobi's Method
120(1)
4.2.7 Gauss-Seidel's Method
121(1)
4.2.7.1 Relaxation
121(1)
4.3 Exercises
122(3)
5 Two-Point Boundary Value Problems
125(22)
5.1 The Finite Element Method (FEM)
125(2)
5.2 Error Estimates in the Energy Norm
127(5)
5.2.1 Adaptivity
132(1)
5.3 FEM for Convection-Diffusion-Absorption BVPs
132(8)
5.4 Exercises
140(7)
6 Scalar Initial Value Problems
147(30)
6.1 Solution Formula and Stability
147(2)
6.2 Finite Difference Methods for IVP
149(2)
6.3 Galerkin Finite Element Methods for IVP
151(5)
6.3.1 The Continuous Galerkin Method
152(2)
6.3.1.1 The cG(2) Algorithm
154(1)
6.3.1.2 The cG(q) Method
154(1)
6.3.2 The Discontinuous Galerkin Method
155(1)
6.4 A Posteriori Error Estimates
156(8)
6.4.1 A Posteriori Error Estimate for cG(1)
156(1)
6.4.1.1 The Dual Problem
157(4)
6.4.2 A Posteriori Error Estimate for dG(0)
161(2)
6.4.3 Adaptivity for dG(0)
163(1)
6.4.3.1 An Adaptivity Algorithm
163(1)
6.5 A Priori Error Analysis
164(4)
6.5.1 A Priori Error Estimates for the dG(0) Method
164(4)
6.6 The Parabolic Case (a(t) > 0)
168(5)
6.6.1 An Example of Error Estimate
171(2)
6.7 Exercises
173(4)
7 Initial Boundary Value Problems in Id
177(30)
7.1 The Heat Equation in Id
177(16)
7.1.1 Stability Estimates
179(4)
7.1.2 FEM for the Heat Equation
183(3)
7.1.3 Error Analysis
186(6)
7.1.4 Exercises
192(1)
7.2 The Wave Equation in Id
193(6)
7.2.1 Wave Equation as a System of Hyperbolic PDEs
194(1)
7.2.2 The Finite Element Discretization Procedure
195(2)
7.2.3 Exercises
197(2)
7.3 Convection-Diffusion Problems
199(8)
7.3.1 Finite Element Method
202(1)
7.3.2 The Streamline-Diffusion Method (SDM)
203(2)
7.3.3 Exercises
205(2)
8 Approximation in Several Dimensions
207(28)
8.1 Introduction
207(2)
8.2 Piecewise Linear Approximation in 2d
209(7)
8.2.1 Basis Functions for the Piecewise Linears in 2d
209(7)
8.3 Constructing Finite Element Spaces
216(3)
8.4 The Interpolant
219(9)
8.4.1 Error Estimates for Piecewise Linear Interpolation
222(6)
8.5 The L2 (Revisited) and Ritz Projections
228(3)
8.5.1 The Ritz or Elliptic Projection
230(1)
8.6 Exercises
231(4)
9 The Boundary Value Problems in UN
235(26)
9.1 The Poisson Equation
235(8)
9.1.1 Weak Stability
236(1)
9.1.2 Error Estimates for the CG(1) FEM
237(5)
9.1.3 Proof of the Regularity Lemma
242(1)
9.2 Stationary Convection-Diffusion Equation
243(6)
9.2.1 The Elliptic Case
243(1)
9.2.1.1 A Brief Note on Distributions
244(4)
9.2.2 Error Estimates
248(1)
9.3 Hyperbolicity Features
249(6)
9.3.1 Convection Dominating Case
250(1)
9.3.2 The SD Method for Convection Diffusion Problem
251(1)
9.3.3 Stability Estimates
252(1)
9.3.4 Error Estimates for Convention Dominating in 2d
252(3)
9.4 Exercises
255(6)
10 The Initial Boundary Value Problems in RN
261(24)
10.1 The Heat Equation in UN
261(11)
10.1.1 The Fundamental Solution
262(1)
10.1.2 Stability
263(2)
10.1.3 The Finite Element for Heat Equation
265(1)
10.1.3.1 The Semidiscrete Problem
265(4)
10.1.4 A Fully Discrete Algorithm
269(1)
10.1.5 The Discrete Equations
270(1)
10.1.6 A Priori Error Estimate: Fully Discrete Problem
271(1)
10.2 The Wave Equation in Rd
272(7)
10.2.1 The Weak Formulation
273(1)
10.2.2 The Semidiscrete Problem
273(1)
10.2.2.1 A Priori Error Estimates for the Semidiscrete Problem
274(1)
10.2.3 The Fully Discrete Problem
275(1)
10.2.3.1 Finite Elements for the Fully Discrete Problem
276(2)
10.2.4 Error Estimate for cG(1)
278(1)
10.3 Exercises
279(6)
Appendix A Answers to Some Exercises
285(10)
Chapter 1 Exercise Section 1.4.1
285(1)
Chapter 1 Exercise Section 1.5.4
285(1)
Chapter 2 Exercise Section 2.11
286(1)
Chapter 3 Exercise Section 3.5
286(1)
Chapter 3 Exercise Section 3.8
287(1)
Chapter 4 Exercise Section 4.3
288(1)
Chapter 5 Exercise Section 5.4
289(2)
Chapter 6 Exercise Section 6.7
291(1)
Chapter 7 Exercise Section 7.2.3
292(1)
Chapter 7 Exercise Section 7.3.3
292(1)
Chapter 9 Poisson Equation. Exercise Section 9.4
292(1)
Chapter 10 IBVPs: Exercise Section 10.3
293(2)
Appendix B Algorithms and Matlab Codes
295(12)
B.1 A Matlab Code to Compute the Mass Matrix M for a Nonuniform Mesh
296(2)
B.1.1 A Matlab Routine to Compute the Load Vector b
297(1)
B.2 Matlab Routine to Compute the L2-Projection
298(2)
B.2.1 A Matlab Routine for the Composite Midpoint Rule
299(1)
B.2.2 A Matlab Routine for the Composite Trapezoidal Rule
299(1)
B.2.3 A Matlab Routine for the Composite Simpson's Rule
299(1)
B.3 A Matlab Routine Assembling the Stiffness Matrix
300(1)
B.4 A Matlab Routine to Assemble the Convection Matrix
301(1)
B.5 Matlab Routine for Forward-, Backward-Euler, and Crank-Nicolson
302(2)
B.6 A Matlab Routine for Mass-Matrix in 2d
304(1)
B.7 A Matlab Routine for a Poisson Assembler in 2d
304(3)
Appendix C Sample Assignments
307(6)
C.1 Assignment 1
307(1)
C.2 Assignment 2
308(5)
C.2.1 Grading Policy of the Assignment
308(1)
C.2.2 Theory
308(1)
C.2.3 Selected Applications
309(1)
C.2.3.1 Convection-Diffusion-Absorption/Reaction
309(1)
C.2.3.2 Electrostatics
310(1)
C.2.3.3 2d Fluid Flow
310(1)
C.2.3 A Heat Conduction
310(1)
C.2.3.5 Quantum Physics
310(3)
Appendix D Symbols
313(4)
D.1 Table of Symbols
313(4)
Bibliography 317(10)
Index 327
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.
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