Introduction to the Finite Element Method for Differential Equations

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.

• Preface xiAcknowledgments xiii1 Introduction 11.1 Preliminaries 21.2 Trinities for Second-Order PDEs 41.3 PDEs in ℝn, Further Classifications 101.4 Differential Operators, Superposition 121.4.1 Exercises 141.5 Some Equations of Mathematical Physics 151.5.1 The Poisson Equation 161.5.2 The Heat Equation 171.5.2.1 A Model Problem for the Stationary Heat Equation in 1d 171.5.2.2 Fourier’s Law of Heat Conduction, Derivation of the Heat Equation 181.5.3 The Wave Equation 211.5.3.1 The Vibrating String, Derivation of the Wave Equation in 1d 211.5.4 Exercises 242 Mathematical Tools 272.1 Vector Spaces 272.1.1 Linear Independence, Basis, and Dimension 302.2 Function Spaces 332.2.1 Spaces of Differentiable Functions 332.2.2 Spaces of Integrable Functions 342.2.3 Weak Derivative 352.2.4 Sobolev Spaces 362.2.5 Hilbert Spaces 372.3 Some Basic Inequalities 382.4 Fundamental Solution of PDEs 412.4.1 Green’s Functions 432.5 The Weak/Variational Formulation 442.6 A Framework for Analytic Solution in 1d 462.6.1 The Variational Formulation in 1d 482.6.2 The Minimization Problem in 1d 512.6.3 A Mixed Boundary Value Problem in 1d 522.7 An Abstract Framework 542.7.1 Riesz and Lax–Milgram Theorems 572.8 Exercises 633 Polynomial Approximation/Interpolation in 1d 673.1 Finite Dimensional Space of Functions on an Interval 673.2 An Ordinary Differential Equation (ODE) 713.2.1 Forward Euler Method to Solve IVP 713.2.2 Variational Formulation for IVP 723.2.3 Galerkin Method for IVP 733.3 A Galerkin Method for (BVP) 743.3.1 An Equivalent Finite Difference Approach 793.4 Exercises 823.5 Polynomial Interpolation in 1d 833.5.1 Lagrange Interpolation 903.6 Orthogonal- and L2-Projection 943.6.1 The L2-Projection onto the Space of Polynomials 943.7 Numerical Integration, Quadrature Rule 963.7.1 Composite Rules for Uniform Partitions 983.7.2 Gauss Quadrature Rule 1013.8 Exercises 1054 Linear Systems of Equations 1094.1 Direct Methods 1104.1.1 LU Factorization of an n × n Matrix A 1134.2 Iterative Methods 1154.2.1 Jacobi Iteration 1154.2.2 Convergence Criterion 1164.2.3 Gauss–Seidel Iteration 1174.2.4 The Successive Over-Relaxation Method (S.O.R.) 1194.2.5 Abstraction of Iterative Methods 1204.2.5.1 Questions 1204.2.6 Jacobi’s Method 1204.2.7 Gauss–Seidel’s Method 1214.2.7.1 Relaxation 1214.3 Exercises 1225 Two-Point Boundary Value Problems 1255.1 The Finite Element Method (FEM) 1255.2 Error Estimates in the Energy Norm 1275.2.1 Adaptivity 1325.3 FEM for Convection–Diffusion–Absorption BVPs 1325.4 Exercises 1406 Scalar Initial Value Problems 1476.1 Solution Formula and Stability 1476.2 Finite Difference Methods for IVP 1496.3 Galerkin Finite Element Methods for IVP 1516.3.1 The Continuous Galerkin Method 1526.3.1.1 The cG(1) Algorithm 1546.3.1.2 The cG(q) Method 1546.3.2 The Discontinuous Galerkin Method 1556.4 A Posteriori Error Estimates 1566.4.1 A Posteriori Error Estimate for cG(1) 1566.4.1.1 The Dual Problem 1576.4.2 A Posteriori Error Estimate for dG(0) 1616.4.3 Adaptivity for dG(0) 1636.4.3.1 An Adaptivity Algorithm 1636.5 A Priori Error Analysis 1646.5.1 A Priori Error Estimates for the dG(0) Method 1646.6 The Parabolic Case (a(t) ≥ 0) 1686.6.1 An Example of Error Estimate 1716.7 Exercises 1737 Initial Boundary Value Problems in 1d 1777.1 The Heat Equation in 1d 1777.1.1 Stability Estimates 1797.1.2 FEM for the Heat Equation 1837.1.3 Error Analysis 1867.1.4 Exercises 1927.2 The Wave Equation in 1d 1937.2.1 Wave Equation as a System of Hyperbolic PDEs 1947.2.2 The Finite Element Discretization Procedure 1957.2.3 Exercises 1977.3 Convection–Diffusion Problems 1997.3.1 Finite Element Method 2027.3.2 The Streamline-Diffusion Method (SDM) 2037.3.3 Exercises 2058 Approximation in Several Dimensions 2078.1 Introduction 2078.2 Piecewise Linear Approximation in 2d 2098.2.1 Basis Functions for the Piecewise Linears in 2d 2098.3 Constructing Finite Element Spaces 2168.4 The Interpolant 2198.4.1 Error Estimates for Piecewise Linear Interpolation 2228.5 The L2(Revisited) and Ritz Projections 2288.5.1 The Ritz or Elliptic Projection 2308.6 Exercises 2319 The Boundary Value Problems in ℝN 2359.1 The Poisson Equation 2359.1.1 Weak Stability 2369.1.2 Error Estimates for the CG(1) FEM 2379.1.3 Proof of the Regularity Lemma 2429.2 Stationary Convection–Diffusion Equation 2439.2.1 The Elliptic Case 2439.2.1.1 A Brief Note on Distributions 2449.2.2 Error Estimates 2489.3 Hyperbolicity Features 2499.3.1 Convection Dominating Case 2509.3.2 The SD Method for Convection Diffusion Problem 2519.3.3 Stability Estimates 2529.3.4 Error Estimates for Convention Dominating in 2d 2529.4 Exercises 25510 The Initial Boundary Value Problems in ℝN 26110.1 The Heat Equation in ℝN26110.1.1 The Fundamental Solution 26210.1.2 Stability 26310.1.3 The Finite Element for Heat Equation 26510.1.3.1 The Semidiscrete Problem 26510.1.4 A Fully Discrete Algorithm 26910.1.5 The Discrete Equations 27010.1.6 A Priori Error Estimate: Fully Discrete Problem 27110.2 The Wave Equation in ℝd 27210.2.1 The Weak Formulation 27310.2.2 The Semidiscrete Problem 27310.2.2.1 A Priori Error Estimates for the Semidiscrete Problem 27410.2.3 The Fully Discrete Problem 27510.2.3.1 Finite Elements for the Fully Discrete Problem 27610.2.4 Error Estimate for cG(1) 27810.3 Exercises 279Appendix A Answers to Some Exercises 285Chapter 1. Exercise Section 1.4.1 285Chapter 1. Exercise Section 1.5.4 285Chapter 2. Exercise Section 2.11 286Chapter 3. Exercise Section 3.5 286Chapter 3. Exercise Section 3.8 287Chapter 4. Exercise Section 4.3 288Chapter 5. Exercise Section 5.4 289Chapter 6. Exercise Section 6.7 291Chapter 7. Exercise Section 7.2.3 292Chapter 7. Exercise Section 7.3.3 292Chapter 9. Poisson Equation. Exercise Section 9.4 292Chapter 10. IBVPs: Exercise Section 10.3 293Appendix B Algorithms and Matlab Codes 295B.1 A Matlab Code to Compute the Mass Matrix M for a Nonuniform Mesh 296B.1.1 A Matlab Routine to Compute the Load Vector b 297B.2 Matlab Routine to Compute the L2-Projection 298B.2.1 A Matlab Routine for the Composite Midpoint Rule 299B.2.2 A Matlab Routine for the Composite Trapezoidal Rule 299B.2.3 A Matlab Routine for the Composite Simpson’s Rule 299B.3 A Matlab Routine Assembling the Stiffness Matrix 300B.4 A Matlab Routine to Assemble the Convection Matrix 301B.5 Matlab Routine for Forward-, Backward-Euler, and Crank–Nicolson 302B.6 A Matlab Routine for Mass-Matrix in 2d 304B.7 A Matlab Routine for a Poisson Assembler in 2d 304Appendix C Sample Assignments 307C.1 Assignment 1 307C.2 Assignment 2 308C.2.1 Grading Policy of the Assignment 308C.2.2 Theory 308C.2.3 Selected Applications 309C.2.3.1 Convection–Diffusion–Absorption/Reaction 309C.2.3.2 Electrostatics 310C.2.3.3 2d Fluid Flow 310C.2.3.4 Heat Conduction 310C.2.3.5 Quantum Physics 310Appendix D Symbols 313D.1 Table of Symbols 313Bibliography 317Index 327

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