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| Preface | |
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| Acknowledgments | |
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| Brief Overview of Partial Differential Equations | |
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| The parabolic equations | |
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| The wave equations | |
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| The elliptic equations | |
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| Differential equations in broader areas | |
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| Electromagnetics | |
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| Fluid mechanics | |
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| Ground water contamination | |
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| Petroleum reservoir simulation | |
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| Finance modeling | |
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| Image processing | |
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| A quick review of numerical methods for PDEs | |
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| References | |
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| Finite Difference Methods for Parabolic Equations | |
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| Introduction | |
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| Theoretical issues: stability, consistence, and convergence | |
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| 1-D parabolic equations | |
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| The [theta]-method | |
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| Some extensions | |
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| 2-D and 3-D parabolic equations | |
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| Standard explicit and implicit methods | |
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| The ADI methods for 2-D problems | |
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| The ADI methods for 3-D problems | |
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| Numerical examples with MATLAB codes | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Finite Difference Methods for Hyperbolic Equations | |
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| Introduction | |
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| Some basic difference schemes | |
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| Dissipation and dispersion errors | |
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| Extensions to conservation laws | |
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| The second-order hyperbolic PDEs | |
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| Numerical examples with MATLAB codes | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Finite Difference Methods for Elliptic Equations | |
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| Introduction | |
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| Numerical solution of linear systems | |
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| Direct methods | |
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| Simple iterative methods | |
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| Modern iterative methods | |
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| Error analysis with a maximum principle | |
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| Some extensions | |
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| Mixed boundary conditions | |
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| Self-adjoint problems | |
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| A fourth-order scheme | |
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| Numerical examples with MATLAB codes | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| High-Order Compact Difference Methods | |
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| One-dimensional problems | |
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| Spatial discretization | |
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| Approximations of high-order derivatives | |
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| Temporal discretization | |
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| Low-pass spatial filter | |
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| Numerical examples with MATLAB codes | |
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| High-dimensional problems | |
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| Temporal discretization for 2-D problems | |
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| Stability analysis | |
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| Extensions to 3-D compact ADI schemes | |
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| Numerical examples with MATLAB codes | |
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| Other high-order compact schemes | |
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| One-dimensional problems | |
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| Two-dimensional problems | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Finite Element Methods: Basic Theory | |
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| Introduction to one-dimensional problems | |
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| The second-order equation | |
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| The fourth-order equation | |
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| Introduction to two-dimensional problems | |
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| The Poisson's equation | |
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| The biharmonic problem | |
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| Abstract finite element theory | |
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| Existence and uniqueness | |
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| Stability and convergence | |
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| Examples of conforming finite element spaces | |
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| Triangular finite elements | |
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| Rectangular finite elements | |
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| Examples of nonconforming finite elements | |
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| Nonconforming triangular elements | |
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| Nonconforming rectangular elements | |
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| Finite element interpolation theory | |
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| Sobolev spaces | |
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| Interpolation theory | |
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| Finite element analysis of elliptic problems | |
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| Analysis of conforming finite elements | |
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| Analysis of nonconforming finite elements | |
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| Finite element analysis of time-dependent problems | |
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| Introduction | |
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| FEM for parabolic equations | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Finite Element Methods: Programming | |
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| FEM mesh generation | |
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| Forming FEM equations | |
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| Calculation of element matrices | |
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| Assembly and implementation of boundary conditions | |
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| The MATLAB code for P[subscript 1] element | |
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| The MATLAB code for the Q[subscript 1] element | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Mixed Finite Element Methods | |
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| An abstract formulation | |
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| Mixed methods for elliptic problems | |
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| The mixed variational formulation | |
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| The mixed finite element spaces | |
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| The error estimates | |
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| Mixed methods for the Stokes problem | |
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| The mixed variational formulation | |
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| Mixed finite element spaces | |
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| An example MATLAB code for the Stokes problem | |
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| Mixed methods for viscous incompressible flows | |
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| The steady Navier-Stokes problem | |
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| The unsteady Navier-Stokes problem | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Finite Element Methods for Electromagnetics | |
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| Introduction to Maxwell's equations | |
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| The time-domain finite element method | |
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| The mixed method | |
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| The standard Galerkin method | |
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| The discontinuous Galerkin method | |
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| The frequency-domain finite element method | |
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| The standard Galerkin method | |
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| The discontinuous Galerkin method | |
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| The mixed DG method | |
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| The Maxwell's equations in dispersive media | |
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| Isotropic cold plasma | |
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| Debye medium | |
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| Lorentz medium | |
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| Double-negative metamaterials | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Meshless Methods with Radial Basis Functions | |
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| Introduction | |
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| The radial basis functions | |
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| The MFS-DRM | |
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| The fundamental solution of PDEs | |
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| The MFS for Laplace's equation | |
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| The MFS-DRM for elliptic equations | |
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| Computing particular solutions using RBFs | |
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| The RBF-MFS | |
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| The MFS-DRM for the parabolic equations | |
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| Kansa's method | |
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| Kansa's method for elliptic problems | |
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| Kansa's method for parabolic equations | |
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| The Hermite-Birkhoff collocation method | |
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| Numerical examples with MATLAB codes | |
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| Elliptic problems | |
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| Biharmonic problems | |
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| Coupling RBF meshless methods with DDM | |
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| Overlapping DDM | |
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| Non-overlapping DDM | |
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| One numerical example | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Other Meshless Methods | |
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| Construction of meshless shape functions | |
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| The smooth particle hydrodynamics method | |
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| The moving least-square approximation | |
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| The partition of unity method | |
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| The element-free Galerkin method | |
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| The meshless local Petrov-Galerkin method | |
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| Bibliographical remarks | |
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| Exercises | |
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| References | |
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| Answers to Selected Problems | |
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| Index | |