Numerical Methods for Exterior Problems

Contents......Page 8
Preface......Page 6
1. Exterior Problems of Partial Differential Equations......Page 11
1.1 Harmonic equation-potential theory......Page 12
1.2 Poisson equations......Page 22
1.3 Poisson equations-variational formulation......Page 23
1.4 Helmholtz equations......Page 27
1.5 Linear elasticity......Page 35
1.6 Bi-harmonic equations......Page 39
1.7.1 Navier-Stokes equations......Page 45
1.7.2 Stokes equations......Page 46
1.7.3 Behavior of solutions at the infinity......Page 49
1.7.5 Oseen flow......Page 51
1.8 Steady Navier-Stokes equations......Page 54
1.9 Heat equation......Page 59
1.10 Wave equation......Page 63
1.11 Maxwell equations......Page 66
1.12 Darwin model......Page 71
2.1.1 Harmonic equation......Page 81
2.1.2 Bi-harmonic equation......Page 85
2.1.3 Stokes equation......Page 87
2.1.4 Plane elasticity......Page 90
2.1.5 Helmholtz equation......Page 92
2.2 General domains......Page 95
2.3 Subdivision of the domain......Page 103
2.4 Dirichlet to Neümann operator......Page 106
2.5 Finite part of divergent integrals......Page 108
2.6 Numerical approximation......Page 113
2.7 Error estimates......Page 118
2.8 Domain decomposition......Page 123
2.9 Boundary perturbation......Page 124
3.1.1 Infinite element formulation......Page 127
3.1.2 Tranfer matrix......Page 130
3.1.3 Further discussion for the transfer matrix......Page 137
3.1.4 Combined stiffness matrix......Page 141
3.2 General elements......Page 143
3.3 Harmonic equation-three dimensional problems......Page 144
3.4 Inhomogeneous equations......Page 146
3.5 Plane elasticity......Page 148
3.6 Bi-harmonic equations......Page 150
3.7 Stokes equation......Page 152
3.8 Darwin model......Page 157
3.9.1 A homogeneous equation......Page 162
3.9.2 An inhomogeneous equation......Page 165
3.9.3 General multiply connected domains......Page 168
3.9.4 Transfer matrices......Page 171
3.10 Convergence......Page 172
4.1 Absorbing boundary conditions......Page 177
4.2 Some approximations......Page 182
4.3 Bayliss-Turkel radiation boundary conditions......Page 185
4.4 A lower order absorbing boundary condition......Page 186
4.6 Maxwell equations......Page 188
4.7 Finite difference schemes......Page 192
4.8.1 Homogeneous boundary condition at the infinity......Page 193
4.8.2 Inhomogeneous boundary conditions at the infinity......Page 196
4.8.3 A linear boundary condition......Page 197
5.1 Wave equations......Page 201
5.2 Bérenger's perfectly matched layers......Page 207
5.3 Stability analysis......Page 211
5.4 Uniaxial perfectly matched layers......Page 218
5.5 Maxwell equations......Page 220
5.6 Helmholtz equations......Page 222
6.1 Introduction......Page 227
6.2 Orthogonal systems of polynomials......Page 235
6.3.1 Mixed Laguerre-Fourier spectral method......Page 240
6.3.2 Spherical harmonic-generalized Laguerre spectral method......Page 245
6.3.3 Generalized Laguerre pseudo-spectral method......Page 247
6.3.4 Nonlinear equations......Page 249
6.4 Jacobi spectral methods......Page 251
6.5 Rational and irrational spectral methods......Page 253
6.6 Error estimates......Page 255
Bibliography......Page 261
Index......Page 275