Numerical Methods for Exterior Problems

✍ Scribed by Long'an Ying

• Publisher: World Scientific
• Year: 2006
• Tongue: English
• Leaves: 280
• Category: Library

✦ Synopsis

This book provides a comprehensive introduction to the numerical methods for the exterior problems in partial differential equations frequently encountered in science and engineering computing. The coverage includes both traditional and novel methods. A concise introduction to the well-posedness of the problems is given, establishing a solid foundation for the methods.

✦ Table of Contents

Contents
Preface
1. Exterior Problems of Partial Differential Equations
1.1 Harmonic equation-potential theory
1.2 Poisson equations
1.3 Poisson equations-variational formulation
1.4 Helmholtz equations
1.5 Linear elasticity
1.6 Bi-harmonic equations
1.7 Steady Navier-Stokes equations -linearized problems
1.7.1 Navier-Stokes equations
1.7.2 Stokes equations
1.7.3 Behavior of solutions at the infinity
1.7.4 Stokes paradox
1.7.5 Oseen flow
1.8 Steady Navier-Stokes equations
1.9 Heat equation
1.10 Wave equation
1.11 Maxwell equations
1.12 Darwin model
2. Boundary Element Method
2.1 Some typical domains
2.1.1 Harmonic equation
2.1.2 Bi-harmonic equation
2.1.3 Stokes equation
2.1.4 Plane elasticity
2.1.5 Helmholtz equation
2.2 General domains
2.3 Subdivision of the domain
2.4 Dirichlet to Neümann operator
2.5 Finite part of divergent integrals
2.6 Numerical approximation
2.7 Error estimates
2.8 Domain decomposition
2.9 Boundary perturbation
3. Infinite Element Method
3.1 Harmonic equation-two dimensional problems
3.1.1 Infinite element formulation
3.1.2 Tranfer matrix
3.1.3 Further discussion for the transfer matrix
3.1.4 Combined stiffness matrix
3.2 General elements
3.3 Harmonic equation-three dimensional problems
3.4 Inhomogeneous equations
3.5 Plane elasticity
3.6 Bi-harmonic equations
3.7 Stokes equation
3.8 Darwin model
3.9 Elliptic equations with variable coefficients
3.9.1 A homogeneous equation
3.9.2 An inhomogeneous equation
3.9.3 General multiply connected domains
3.9.4 Transfer matrices
3.10 Convergence
4. Artificial Boundary Conditions
4.1 Absorbing boundary conditions
4.2 Some approximations
4.3 Bayliss-Turkel radiation boundary conditions
4.4 A lower order absorbing boundary condition
4.5 Liao extrapolation in space and time
4.6 Maxwell equations
4.7 Finite difference schemes
4.8 Stationary Navier-Stokes equations
4.8.1 Homogeneous boundary condition at the infinity
4.8.2 Inhomogeneous boundary conditions at the infinity
4.8.3 A linear boundary condition
5. Perfectly Matched Layer Method
5.1 Wave equations
5.2 Bérenger's perfectly matched layers
5.3 Stability analysis
5.4 Uniaxial perfectly matched layers
5.5 Maxwell equations
5.6 Helmholtz equations
6. Spectral Method
6.1 Introduction
6.2 Orthogonal systems of polynomials
6.3 Laguerre spectral methods
6.3.1 Mixed Laguerre-Fourier spectral method
6.3.2 Spherical harmonic-generalized Laguerre spectral method
6.3.3 Generalized Laguerre pseudo-spectral method
6.3.4 Nonlinear equations
6.4 Jacobi spectral methods
6.5 Rational and irrational spectral methods
6.6 Error estimates
Bibliography
Index