1 Introduction 1.1 Background 1.2 Notation and Mathematical Preliminaries 1.3 Physical Derivation of the Heat Equation 1.4 Problems 2 A Two-Point Boundary Value Problem 2.1 The Maximum Principle 2.2 Green's Function 2.3 Variational Formulation 2.4 Problems 3 Elliptic Equations 3.1 Preliminaries 3.2 A Maximum Principle 3.3 Dirichlet's Problem for a Disc. Poisson's Integral 3.4 Fundamental Solutions. Green's Function 3.5 Variational Formulation of the Dirichlet Problem 3.6 A Neumann Problem 3.7 Regularity 3.8 Problems 4 Finite Difference Methods for Elliptic Equations 4.1 A Two-Point Boundary Value Problem 4.2 Poisson's Equation 4.3 Problems 5 Finite Element Methods for Elliptic Equations 5.1 A Two-Point boundary Value Problem 5.2 A Model Problem in the Plane 5.3 Some Facts from Approximation Theory 5.4 Error Estimates 5.5 An A Posteriori Error Estimate 5.6 Numerical Integration 5.7 A Mixed Finite Element Method 5.8 Problems 6 The Elliptic Eigenvalue Problem 6.1 Eigenfunction Expansions 6.2 Numerical Solution of the Eigenvalue Problem 6.3 Problems 7 Initial-Value Problems for ODEs 7.1 The Initial Value Problem for a Linear System 7.2 Numerical Solution of ODEs 7.3 Problems 8 Parabolic Equations 8.1 The Pure Initial Value Problem 8.2 Solution by Eigenfunction Expansion 8.3 Variational Formulation. Energy Estimates 8.4 A Maximum Principle 8.5 Problems 9 Finite Difference Methods for Parabolic Problems 9.1 The Pure Initial Value Problem 9.2 The Mixed Initial-Boundary Value Problem 9.3 Problems 10 The Finite Element Method for a Parabolic Problem 10.1 The Semidiscrete Galerkin Finite Element Method 10.2 Some Completely Discrete Schemes 10.3 Problems 11 Hyperbolic Equations 11.1 Characteristic Directions and Surfaces 11.2 The Wave Equation 11.3 First Order Scalar Equation 11.4 Symmetric Hyperbolic Systems 11.5 Problems 12 Finite Difference Methods for Hyperbolic Equations 12.1 First Order Scalar Equations 12.2 Symmetric Hyperbolic Systems 12.3 The Wendroff Box Scheme 12.4 Problems 13 The Finite Element Method for Hyperbolic Equations 13.1 The Wave Equation 13.2 First Order Hyperbolic Equations 13.3 Problems 14 Some Other Classes of Numerical Methods 14.1 Collocation Methods 14.2 Spectral Methods 14.3 Finite Volume Methods 14.4 Boundary Element Methods 14.5 Problems A Some Tools from Mathematical Analysis A.1 Abstract Linear Spaces A.2 Function Spaces A.3 The Fourier Transform A.4 Problems B Orientation on Numerical Linear Algebra B.1 Direct Methods B.2 Iterative Methods. Relaxation, Overrelaxation, and Acceleration B.3 Preconditioned Conjugate Gradient Methods B.4 Preconditioned Conjugate Gradient Methods B.5 Multigrid and Domain Decomposition Methods Bibliography Index
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