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**Unformatted text preview: **MATH 8445, University of Minnesota, Fall 2009 Numerical Analysis of Differential Equations Instructors notes Douglas N. Arnold c 2009 by Douglas N. Arnold. These notes may not be duplicated without explicit permission from the author. Contents Chapter 1. Introduction 1 1. Basic examples of PDEs 1 1.1. Heat flow and the heat equation 1 1.2. Elastic membranes 3 1.3. Elastic plates 3 2. Some motivations for studying the numerical analysis of PDE 4 Chapter 2. The finite difference method for the Laplacian 7 1. The 5-point difference operator 7 2. Analysis via a maximum principle 10 3. Consistency, stability, and convergence 11 4. Fourier analysis 13 5. Analysis via summation by parts 15 6. Extensions 17 6.1. Curved boundaries 17 6.2. More general PDEs 20 6.3. More general boundary conditions 21 6.4. Nonlinear problems 21 Chapter 3. Linear algebraic solvers 23 1. Classical iterations 23 2. The conjugate gradient method 29 2.1. Line search methods and the method of steepest descents 29 2.2. The conjugate gradient method 31 2.3. Preconditioning 37 3. Multigrid methods 39 Chapter 4. Finite element methods for elliptic equations 49 1. Weak and variational formulations 49 2. Galerkin method and finite elements 50 3. Lagrange finite elements 51 4. Coercivity, inf-sup condition, and well-posedness 53 4.1. The symmetric coercive case 54 4.2. The coercive case 55 4.3. The inf-sup condition 55 5. Stability, consistency, and convergence 56 6. Finite element approximation theory 57 7. Error estimates for finite elements 62 3 4 CONTENTS 7.1. Estimate in H 1 62 7.2. Estimate in L 2 63 8. A posteriori error estimates and adaptivity 64 8.1. The Cl ement interpolant 64 8.2. The residual and the error 67 8.3. Estimating the residual 68 8.4. A posteriori error indicators 69 8.5. Examples of adaptive finite element computations 70 Chapter 5. Time-dependent problems 73 1. Finite difference methods for the heat equation 73 1.1. Forward differences in time 74 1.2. Backward differences in time 76 1.3. Crank-Nicolson 77 1.4. Fourier analysis 77 2. Finite element methods for the heat equation 78 2.1. Analysis of the semidiscrete finite element method 79 2.2. Analysis of a fully discrete finite element method 81 CHAPTER 1 Introduction Galileo wrote that the great book of nature is written in the language of mathemat- ics. The most precise and concise description of many physical systems is through partial differential equations. 1. Basic examples of PDEs 1.1. Heat flow and the heat equation. We start with a typical physical application of partial differential equations, the modeling of heat flow. Suppose we have a solid body occupying a region R 3 . The temperature distribution in the body can be given by a function u : J R where J is an interval of time we are interested in and u ( x,t ) is the temperature at a point x at time t J . The heat content (the amount of thermal energy) in a subbody D is given by heat content of D = Z D cudx where...

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