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| Introduction | |
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| Background | |
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| Notation and Mathematical Preliminaries | |
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| Physical Derivation of the Heat Equation | |
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| Problems | |
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| A Two-Point Boundary Value Problem | |
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| The Maximum Principle | |
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| Green's Function | |
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| Variational Formulation | |
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| Problems | |
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| Elliptic Equations | |
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| Preliminaries | |
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| A Maximum Principle | |
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| Dirichlet's Problem for a Disc. Poisson's Integral | |
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| Fundamental Solutions. Green's Function | |
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| Variational Formulation of the Dirichlet Problem | |
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| A Neumann Problem | |
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| Regularity | |
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| Problems | |
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| Finite Difference Methods for Elliptic Equations | |
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| A Two-Point Boundary Value Problem | |
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| Poisson's Equation | |
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| Problems | |
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| Finite Element Methods for Elliptic Equations | |
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| A Two-Point Boundary Value Problem | |
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| A Model Problem in the Plane | |
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| Some Facts from Approximation Theory | |
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| Error Estimates | |
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| An A Posteriori Error Estimate | |
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| Numerical Integration | |
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| A Mixed Finite Element Method | |
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| Problems | |
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| The Elliptic Eigenvalue Problem | |
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| Eigenfunction Expansions | |
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| Numerical Solution of the Eigenvalue Problem | |
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| Problems | |
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| Initial-Value Problems for ODEs | |
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| The Initial Value Problem for a Linear System | |
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| Numerical Solution of ODEs | |
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| Problems | |
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| Parabolic Equations | |
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| The Pure Initial Value Problem | |
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| Solution by Eigenfunction Expansion | |
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| Variational Formulation. Energy Estimates | |
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| A Maximum Principle | |
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| Problems | |
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| Finite Difference Methods for Parabolic Problems | |
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| The Pure Initial Value Problem | |
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| The Mixed Initial-Boundary Value Problem | |
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| Problems | |
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| The Finite Element Method for a Parabolic Problem | |
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| The Semidiscrete Galerkin Finite Element Method | |
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| Some Completely Discrete Schemes | |
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| Problems | |
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| Hyperbolic Equations | |
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| Characteristic Directions and Surfaces | |
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| The Wave Equation | |
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| First Order Scalar Equations | |
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| Symmetric Hyperbolic Systems | |
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| Problems | |
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| Finite Difference Methods for Hyperbolic Equations | |
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| First Order Scalar Equations | |
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| Symmetric Hyperbolic Systems | |
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| The Wendroff Box Scheme | |
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| Problems | |
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| The Finite Element Method for Hyperbolic Equations | |
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| The Wave Equation | |
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| First Order Hyperbolic Equations | |
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| Problems | |
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| Some Other Classes of Numerical Methods | |
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| Collocation methods | |
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| Spectral Methods | |
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| Finite Volume Methods | |
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| Boundary Element Methods | |
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| Problems | |
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| Some Tools from Mathematical Analysis | |
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| Abstract Linear Spaces | |
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| Function Spaces | |
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| The Fourier Transform | |
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| Problems | |
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| Orientation on Numerical Linear Algebra | |
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| Direct Methods | |
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| Iterative Methods. Relaxation, Overrelaxation, and Acceleration | |
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| Alternating Direction Methods | |
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| Preconditioned Conjugate Gradient Methods | |
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| Multigrid and Domain Decomposition Methods | |
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| Bibliography | |
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| Index | |