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An Introduction to the Finite Element Method for Differential Equations


An Introduction to the Finite Element Method for Differential Equations


1. Aufl.

von: Mohammad Asadzadeh

88,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 27.08.2020
ISBN/EAN: 9781119671664
Sprache: englisch
Anzahl Seiten: 352

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Beschreibungen

<p><b>Master the finite element method with this masterful and practical volume</b></p> <p><i>An Introduction to the Finite Element Method (FEM) for Differential Equations</i> 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.</p> <p>The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, <i>An Introduction to the Finite Element Method</i> covers topics including:</p> <ul> <li>An introduction to basic ordinary and partial differential equations</li> <li>The concept of fundamental solutions using Green's function approaches</li> <li>Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations</li> <li>Higher-dimensional interpolation procedures</li> <li>Stability and convergence analysis of FEM for differential equations</li> </ul> <p>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.</p>
<p>Preface <i>xi</i></p> <p>Acknowledgments <i>xiii</i></p> <p><b>1 Introduction </b><b>1</b></p> <p>1.1 Preliminaries 2</p> <p>1.2 Trinities for Second-Order PDEs 4</p> <p>1.3 PDEs in ℝ<i><sup>n</sup></i>, Further Classifications 10</p> <p>1.4 Differential Operators, Superposition 12</p> <p>1.4.1 Exercises 14</p> <p>1.5 Some Equations of Mathematical Physics 15</p> <p>1.5.1 The Poisson Equation 16</p> <p>1.5.2 The Heat Equation 17</p> <p>1.5.2.1 A Model Problem for the Stationary Heat Equation in 1<i>d </i>17</p> <p>1.5.2.2 Fourier’s Law of Heat Conduction, Derivation of the Heat Equation 18</p> <p>1.5.3 The Wave Equation 21</p> <p>1.5.3.1 The Vibrating String, Derivation of the Wave Equation in 1<i>d </i>21</p> <p>1.5.4 Exercises 24</p> <p><b>2 Mathematical Tools </b><b>27</b></p> <p>2.1 Vector Spaces 27</p> <p>2.1.1 Linear Independence, Basis, and Dimension 30</p> <p>2.2 Function Spaces 33</p> <p>2.2.1 Spaces of Differentiable Functions 33</p> <p>2.2.2 Spaces of Integrable Functions 34</p> <p>2.2.3 Weak Derivative 35</p> <p>2.2.4 Sobolev Spaces 36</p> <p>2.2.5 Hilbert Spaces 37</p> <p>2.3 Some Basic Inequalities 38</p> <p>2.4 Fundamental Solution of PDEs 41</p> <p>2.4.1 Green’s Functions 43</p> <p>2.5 The Weak/Variational Formulation 44</p> <p>2.6 A Framework for Analytic Solution in 1<i>d </i>46</p> <p>2.6.1 The Variational Formulation in 1<i>d </i>48</p> <p>2.6.2 The Minimization Problem in 1<i>d </i>51</p> <p>2.6.3 A Mixed Boundary Value Problem in 1<i>d </i>52</p> <p>2.7 An Abstract Framework 54</p> <p>2.7.1 Riesz and Lax–Milgram Theorems 57</p> <p>2.8 Exercises 63</p> <p><b>3 Polynomial Approximation/Interpolation in 1<i>d </i></b><b>67</b></p> <p>3.1 Finite Dimensional Space of Functions on an Interval 67</p> <p>3.2 An Ordinary Differential Equation (ODE) 71</p> <p>3.2.1 Forward Euler Method to Solve IVP 71</p> <p>3.2.2 Variational Formulation for IVP 72</p> <p>3.2.3 Galerkin Method for IVP 73</p> <p>3.3 A Galerkin Method for (BVP) 74</p> <p>3.3.1 An Equivalent Finite Difference Approach 79</p> <p>3.4 Exercises 82</p> <p>3.5 Polynomial Interpolation in 1<i>d </i>83</p> <p>3.5.1 Lagrange Interpolation 90</p> <p>3.6 Orthogonal- and <i>L</i><sub>2</sub>-Projection 94</p> <p>3.6.1 The <i>L</i><sub>2</sub>-Projection onto the Space of Polynomials 94</p> <p>3.7 Numerical Integration, Quadrature Rule 96</p> <p>3.7.1 Composite Rules for Uniform Partitions 98</p> <p>3.7.2 Gauss Quadrature Rule 101</p> <p>3.8 Exercises 105</p> <p><b>4 Linear Systems of Equations </b><b>109</b></p> <p>4.1 Direct Methods 110</p> <p>4.1.1 LU Factorization of an <i>n </i>× <i>n </i>Matrix <b>A </b>113</p> <p>4.2 Iterative Methods 115</p> <p>4.2.1 Jacobi Iteration 115</p> <p>4.2.2 Convergence Criterion 116</p> <p>4.2.3 Gauss–Seidel Iteration 117</p> <p>4.2.4 The Successive Over-Relaxation Method (S.O.R.) 119</p> <p>4.2.5 Abstraction of Iterative Methods 120</p> <p>4.2.5.1 Questions 120</p> <p>4.2.6 Jacobi’s Method 120</p> <p>4.2.7 Gauss–Seidel’s Method 121</p> <p>4.2.7.1 Relaxation 121</p> <p>4.3 Exercises 122</p> <p><b>5 Two-Point Boundary Value Problems </b><b>125</b></p> <p>5.1 The Finite Element Method (FEM) 125</p> <p>5.2 Error Estimates in the Energy Norm 127</p> <p>5.2.1 Adaptivity 132</p> <p>5.3 FEM for Convection–Diffusion–Absorption BVPs 132</p> <p>5.4 Exercises 140</p> <p><b>6 Scalar Initial Value Problems </b><b>147</b></p> <p>6.1 Solution Formula and Stability 147</p> <p>6.2 Finite Difference Methods for IVP 149</p> <p>6.3 Galerkin Finite Element Methods for IVP 151</p> <p>6.3.1 The Continuous Galerkin Method 152</p> <p>6.3.1.1 The cG(1) Algorithm 154</p> <p>6.3.1.2 The cG(<i>q</i>) Method 154</p> <p>6.3.2 The Discontinuous Galerkin Method 155</p> <p>6.4 A Posteriori Error Estimates 156</p> <p>6.4.1 A Posteriori Error Estimate for cG(1) 156</p> <p>6.4.1.1 The Dual Problem 157</p> <p>6.4.2 A Posteriori Error Estimate for dG(0) 161</p> <p>6.4.3 Adaptivity for dG(0) 163</p> <p>6.4.3.1 An Adaptivity Algorithm 163</p> <p>6.5 A Priori Error Analysis 164</p> <p>6.5.1 A Priori Error Estimates for the dG(0) Method 164</p> <p>6.6 The Parabolic Case (<i>a</i>(<i>t</i>) ≥ 0) 168</p> <p>6.6.1 An Example of Error Estimate 171</p> <p>6.7 Exercises 173</p> <p><b>7 Initial Boundary Value Problems in 1<i>d </i></b><b>177</b></p> <p>7.1 The Heat Equation in 1<i>d </i>177</p> <p>7.1.1 Stability Estimates 179</p> <p>7.1.2 FEM for the Heat Equation 183</p> <p>7.1.3 Error Analysis 186</p> <p>7.1.4 Exercises 192</p> <p>7.2 The Wave Equation in 1<i>d </i>193</p> <p>7.2.1 Wave Equation as a System of Hyperbolic PDEs 194</p> <p>7.2.2 The Finite Element Discretization Procedure 195</p> <p>7.2.3 Exercises 197</p> <p>7.3 Convection–Diffusion Problems 199</p> <p>7.3.1 Finite Element Method 202</p> <p>7.3.2 The Streamline-Diffusion Method (SDM) 203</p> <p>7.3.3 Exercises 205</p> <p><b>8 Approximation in Several Dimensions </b><b>207</b></p> <p>8.1 Introduction 207</p> <p>8.2 Piecewise Linear Approximation in 2<i>d </i>209</p> <p>8.2.1 Basis Functions for the Piecewise Linears in 2<i>d </i>209</p> <p>8.3 Constructing Finite Element Spaces 216</p> <p>8.4 The Interpolant 219</p> <p>8.4.1 Error Estimates for Piecewise Linear Interpolation 222</p> <p>8.5 The <i>L</i><sub>2</sub> (Revisited) and Ritz Projections 228</p> <p>8.5.1 The Ritz or Elliptic Projection 230</p> <p>8.6 Exercises 231</p> <p><b>9 The Boundary Value Problems in </b><b>ℝ<i><sup>N</sup> </i></b><b>235</b></p> <p>9.1 The Poisson Equation 235</p> <p>9.1.1 Weak Stability 236</p> <p>9.1.2 Error Estimates for the CG(1) FEM 237</p> <p>9.1.3 Proof of the Regularity Lemma 242</p> <p>9.2 Stationary Convection–Diffusion Equation 243</p> <p>9.2.1 The Elliptic Case 243</p> <p>9.2.1.1 A Brief Note on Distributions 244</p> <p>9.2.2 Error Estimates 248</p> <p>9.3 Hyperbolicity Features 249</p> <p>9.3.1 Convection Dominating Case 250</p> <p>9.3.2 The SD Method for Convection Diffusion Problem 251</p> <p>9.3.3 Stability Estimates 252</p> <p>9.3.4 Error Estimates for Convention Dominating in 2<i>d </i>252</p> <p>9.4 Exercises 255</p> <p><b>10 The Initial Boundary Value Problems in </b><b>ℝ<i><sup>N</sup> </i></b><b>261</b></p> <p>10.1 The Heat Equation in ℝ<b><i><sup>N</sup></i></b>261</p> <p>10.1.1 The Fundamental Solution 262</p> <p>10.1.2 Stability 263</p> <p>10.1.3 The Finite Element for Heat Equation 265</p> <p>10.1.3.1 The Semidiscrete Problem 265</p> <p>10.1.4 A Fully Discrete Algorithm 269</p> <p>10.1.5 The Discrete Equations 270</p> <p>10.1.6 A Priori Error Estimate: Fully Discrete Problem 271</p> <p>10.2 The Wave Equation in ℝ<i><sup>d</sup> </i>272</p> <p>10.2.1 The Weak Formulation 273</p> <p>10.2.2 The Semidiscrete Problem 273</p> <p>10.2.2.1 A Priori Error Estimates for the Semidiscrete Problem 274</p> <p>10.2.3 The Fully Discrete Problem 275</p> <p>10.2.3.1 Finite Elements for the Fully Discrete Problem 276</p> <p>10.2.4 Error Estimate for cG(1) 278</p> <p>10.3 Exercises 279</p> <p><b>Appendix A Answers to Some Exercises </b><b>285</b></p> <p>Chapter 1. Exercise Section 1.4.1 285</p> <p>Chapter 1. Exercise Section 1.5.4 285</p> <p>Chapter 2. Exercise Section 2.11 286</p> <p>Chapter 3. Exercise Section 3.5 286</p> <p>Chapter 3. Exercise Section 3.8 287</p> <p>Chapter 4. Exercise Section 4.3 288</p> <p>Chapter 5. Exercise Section 5.4 289</p> <p>Chapter 6. Exercise Section 6.7 291</p> <p>Chapter 7. Exercise Section 7.2.3 292</p> <p>Chapter 7. Exercise Section 7.3.3 292</p> <p>Chapter 9. Poisson Equation. Exercise Section 9.4 292</p> <p>Chapter 10. IBVPs: Exercise Section 10.3 293</p> <p><b>Appendix B Algorithms and Matlab Codes </b><b>295</b></p> <p>B.1 A Matlab Code to Compute the Mass Matrix <b>M </b>for a Nonuniform Mesh 296</p> <p>B.1.1 A Matlab Routine to Compute the Load Vector <b>b </b>297</p> <p>B.2 Matlab Routine to Compute the <i>L</i><sub>2</sub>-Projection 298</p> <p>B.2.1 A Matlab Routine for the Composite Midpoint Rule 299</p> <p>B.2.2 A Matlab Routine for the Composite Trapezoidal Rule 299</p> <p>B.2.3 A Matlab Routine for the Composite Simpson’s Rule 299</p> <p>B.3 A Matlab Routine Assembling the Stiffness Matrix 300</p> <p>B.4 A Matlab Routine to Assemble the Convection Matrix 301</p> <p>B.5 Matlab Routine for Forward-, Backward-Euler, and Crank–Nicolson 302</p> <p>B.6 A Matlab Routine for Mass-Matrix in 2<i>d </i>304</p> <p>B.7 A Matlab Routine for a Poisson Assembler in 2<i>d </i>304</p> <p><b>Appendix C Sample Assignments </b><b>307</b></p> <p>C.1 Assignment 1 307</p> <p>C.2 Assignment 2 308</p> <p>C.2.1 Grading Policy of the Assignment 308</p> <p>C.2.2 Theory 308</p> <p>C.2.3 Selected Applications 309</p> <p>C.2.3.1 Convection–Diffusion–Absorption/Reaction 309</p> <p>C.2.3.2 Electrostatics 310</p> <p>C.2.3.3 2<i>d </i>Fluid Flow 310</p> <p>C.2.3.4 Heat Conduction 310</p> <p>C.2.3.5 Quantum Physics 310</p> <p><b>Appendix D Symbols </b><b>313</b></p> <p>D.1 Table of Symbols 313</p> <p>Bibliography 317</p> <p>Index 327</p>
<p><b>MOHAMMAD ASADZADEH, P<small>H</small>D</b> 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.
<p><b>Master the finite element method with this masterful and practical volume</b> <p><i>An Introduction to the Finite Element Method (FEM) for Differential Equations</i> 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. <p>The book is filled with concrete strategies and useful methods to simplify its complex mathematical contents. Practically written and carefully detailed, <i>An Introduction to the Finite Element Method</i> covers topics including: <ul> <li>An introduction to basic ordinary and partial differential equations</li> <li>The concept of fundamental solutions using Green's function approaches</li> <li>Polynomial approximations and interpolations, quadrature rules, and iterative numerical methods to solve linear systems of equations</li> <li>Higher-dimensional interpolation procedures</li> <li>Stability and convergence analysis of FEM for differential equations</li> </ul> <p>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.

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