Contents
Introduction......................................................................................................1 References ...................................................................................................5
Chapter 1 Global ABCs for Second Order Elliptic Equations.......................9
1.1 Exterior Problem of Second Order Elliptic Equations .......................9
1.2 Global ABCs for the Exterior Problem of 2-D Poisson Equation...............................................................................13
1.2.1 Steklov-Poincar¨¦ Mapping for the Exterior Problem of Laplace Equation ..............................................................14
1.2.2 The Reduced Boundary Value Problem on .i.......................17
1.2.3 Finite Element Approximation of the Reduced Boundary Value Problem (1.2.30)~(1.2.32)...........................21
1.3 Global ABCs for the Exterior Problems of 3-D Poisson Equation...............................................................................26
1.3.1 Exact and Approximate ABCs on the Spherical Artificial Boundary ¦£R...........................................................26
1.3.2 Equivalent and Approximate Boundary Value Problems on the Bounded Computational Domain .i ..........................30
1.3.3 Finite Element Approximation of the Variational Problem (1.3.30)....................................................................34
1.4 Exterior Problem of the Modified Helmholtz Equation....................37
1.4.1 Global Boundary Condition of the Exterior Problem for the 2-D Modified Helmholtz Equation.............................37
1.4.2 The Reduced Boundary Value Problem on the Computational Domain .i.....................................................39
1.4.3 Finite Element Approximation of the Reduced Boundary Value Problem......................................................45
1.4.4 Global Boundary Condition of the Exterior Problem for the 3-D Modified Helmholtz Equation.............................47
1.5 Global ABCs for the Exterior Problems of the Helmholtz Equation..........................................................................49
1.5.1 Dirichlet to Sommerfeld Mapping of the Exterior Problem of the 2-D Helmholtz Equation...............................49
1.5.2 Dirichlet to Sommerfeld Mapping of the Exterior Problem of the 3-D Helmholtz Equation...............................55
References .................................................................................................58
Chapter 2 Global ABCs for the Navier System and Stokes System............61
2.1 Navier System and Stokes System....................................................61
2.2 The Exterior Problem of the 2-D Navier System.............................64
2.2.1 The Global Boundary Condition on the Artificial Boundary ¦£R...........................................................65
2.2.2 The Reduced Problem on the Bounded Domain...................71
2.2.3 The Finite Element Approximation for the Reduced Problem (2.2.59)....................................................................77
2.3 Exterior Problem of the 2-D Stokes System.....................................79
2.3.1 Highly Accurate Approximate Artificial Boundary Condition..............................................................80
2.3.2 Finite Element Approximation on the Computational Domain .i for the Reduced Problem ....................................84
2.4 Vector Fields on the Spherical Surface.............................................91
2.5 Global ABCs for the Exterior Problem of 3-D Navier System...........96
2.5.1 Highly Accurate Approximate ABCs....................................96
2.5.2 Finite Element Approximation of the Variational Problem on the Bounded Computational Domain .i ........... 100 References ............................................................................................... 111
Chapter 3 Global ABCs for Heat and Schr.dinger Equations................... 115
3.1 Heat Equations on Unbounded Domains........................................ 115
3.2 1-D Heat Equations on Unbounded Domains................................. 117
3.2.1 Exact Boundary Conditions on the Artificial Boundary ¦²0 ........................................................................ 117
3.2.2 Finite Difference Approximation for the Reduced Problem (3.2.7)~(3.2.10) ..................................................... 119
3.2.3 Stability Analysis of Scheme (3.2.29)~(3.2.33)....................126
3.3 Global Boundary Conditions for Exterior Problems of 2-D Heat Equations ........................................................................ 131
3.3.1 Exact and Approximate Conditions on the Artificial Boundary ¦²R......................................................... 132
3.3.2 Finite Difference Approximation of the Reduced Problem (3.3.37)~(3.3.40) ................................................... 138
3.4 Global Boundary Conditions for Exterior Problems of 3-D Heat Equations ........................................................................ 140
3.4.1 Exact and Approximate Conditions on the Artificial Boundary ¦²R......................................................... 140
3.4.2 Stability Analysis for the Reduced Initial Boundary Value Problem.....................................................................147
3.4.3 The Finite Element Approximation for the Reduced
Initial Boundary Value Problem (3.4.38)~(3.4.41)..............150
3.5 Schr.dinger Equation on Unbounded Domains..............................151
3.6 1-D Schr.dinger Equation on Unbounded Domains.......................152
3.6.1 The Reduced Initial Value Problem and its Finite Difference Approximation ................................................... 153
3.6.2 Stability and Convergence Analysis of Scheme (3.6.19)~(3.6.22).................................................................. 158
3.7 The Global Boundary Condition for the Exterior Problem of the 2-D Linear Schr.dinger Equation.........................................166
3.7.1 Exact and Approximate Boundary Conditions on the Artificial Boundary ¦²R ............................................. 167
3.7.2 Stability Analysis of the Reduced Approximate Initial Boundary Value Problem .................................................... 172
3.8 The Global Boundary Condition for the Exterior Problem of the 3-D Linear Schr.dinger Equation.........................................175
3.8.1 Exact and Approximate Boundary Conditions on the Artificial Boundary ¦²R ............................................. 176
3.8.2 Stability Analysis of the Reduced Approximate Initial
Boundary Value Problem .................................................... 183 References ............................................................................................... 187
Chapter 4 ABCs for Wave Equation, Klein-Gordon Equation, and Linear KdV Equations..................................................................... 189
4.1 1-D Wave Equation......................................................................... 189
4.1.1 Transparent Boundary Conditions on the Artificial Boundaries ¦²1 and ¦²0 ........................................... 190
4.2 2-D Wave Equation......................................................................... 192
4.2.1 Absorbing Boundary Conditions ......................................... 193
4.2.2 The Initial Boundary Value Problem on the Bounded Computational Domain Di ................................... 200
4.3 3-D Wave Equation......................................................................... 203
4.3.1 Absorbing Boundary Condition on the Artificial Boundary ¦²R......................................................... 204
4.3.2 The Equivalent and Approximate Initial Boundary Value Problem on the Bounded Computational Domain Di ........... 208
4.4 1-D Klein-Gordon Equation............................................................ 209
4.4.1 Absorbing Boundary Conditions on the Artificial Boundary ¦²1, ¦²0................................................................... 210
4.4.2 The Initial Boundary Value Problem on the Bounded Computational Domain Di .................................................. 212
4.5 2- and 3-D Klein-Gordon Equations...............................................214
4.5.1 Absorbing Boundary Conditions on the Artificial Boundary ¦²R (2-D case) ...................................................... 215
4.5.2 Absorbing Boundary Conditions on the Artificial Boundary ¦²R (3-D case) ...................................................... 220
4.5.3 The Initial Boundary Value Problem on the Bounded Computational Domain Di .................................................. 223
4.6 Linear KdV Equation ..................................................................... 224
4.6.1 Absorbing Boundary Condition on the Artificial Boundaries ¦²a and ¦²b........................................................... 225
4.6.2 The Equivalent Initial Boundary Value Problem on the Bounded Computational Domain.................................. 227
4.7 Appendix: Three Integration Formulas .......................................... 228 References ............................................................................................... 232
Chapter 5 Local Artificial Boundary Conditions ....................................... 233
5.1 Local Boundary Conditions for Exterior Problems of the 2-D Poisson Equation ............................................................... 234
5.1.1 Local Boundary Condition on the Artificial Bboundary ¦£R ...................................................................... 234
5.1.2 Finite Element Approximation Using the Local Boundary Condition and its Error Estimate....................... 236
5.2 Local Boundary Conditions for the 3-D Poisson Equation............. 241
5.2.1 The Local Boundary Condition on the Artificial Boundary ¦£R for Problem (I)............................................... 242
5.2.2 Local Boundary Conditions on the Artificial Boundary ¦£R for Problem (II) ............................................. 250
5.3 Local ABCs for Wave Equations on Unbounded Domains............. 254 References ............................................................................................... 257
Chapter 6 Discrete Artificial Boundary Conditions................................... 259
6.1 Boundary Condition on a Polygon Boundary for the 2-D Poisson Equation¡ªThe Method of Lines.......................................260
6.1.1 Discrete Boundary Conditions on Polygonal Boundaries........................................................................... 260
6.1.2 Numerical Approximation of the Exterior Problem (6.1.1)~(6.1.3)...................................................................... 268
6.2 2-D Viscous Incompressible Flow in a Channel¡ªInfinite Difference Method........................................................................... 270
6.2.1 2-D Viscous Incompressible Flow in a Channel................... 270
6.2.2 Discrete ABCs ..................................................................... 272
6.3 Numerical Simulation of Infinite Elastic Foundation¡ªInfinite Element Method.............................................................................. 278
6.3.1 The Steklov-Poincar¨¨ on an Artificial Boundary of Line Segments ..................................................................... 279
6.3.2 Numerical Approximation for the Bilinear Form B(u, v)........................................................................ 281
6.3.3 A Direct Method for Solving the Infinite System of Algebraic Equations (6.3.25)...........................................284
6.3.4 A Fast Iteration Method for Computing the Combined Stiffness Matrix KZ............................................. 289
6.4 Discrete Absorbing Boundary Condition for the 1-D Klein-Gordon Equation¡ªZ transform method .............................. 292
6.4.1 Z Transform......................................................................... 292
6.4.2 Discrete Absorbing ABC ..................................................... 294
6.4.3 Finite Difference Approximation for the 1-D Klein-Gordon Equation on the Bounded Domain...............296 References ............................................................................................... 297
Chapter 7 Implicit Artificial Boundary Conditions ................................... 299
7.1 Implicit Boundary Condition for the Exterior Problem of the 2-D Poisson Equation ............................................................... 300
7.1.1 The Single and Double Layer Potential, and Their Derivative for the 2-D Laplace Equation ............................ 300
7.1.2 The Derivation of the Implicit ABC for the Exterior Problem of the 2-D Poisson Equation.................................305
7.1.3 The Finite Element Approximation and Error Estimate for the Variational Problem (7.1.37) ................................... 309
7.2 Implicit Boundary Condition for the Exterior Problem of the 3-D Poisson Equation ............................................................... 310
7.3 ABC for the ExteriorProblem of the Helmholtz Equation............316
7.3.1 The Normal Derivative on ¦£A for the Double Layer Potential of the Helmholtz Equation................................... 318
7.4 Implicit ABCs for the Exterior Problems of the Navier System.................................................................................321
7.4.1 Fundamental Solution, Stress Operator, Single and Double Layer Potentials ...................................................... 321
7.4.2 New Forms of T(.x, nx)vII (x) on ¦£A (n = 2) ....................... 323
7.4.3 New Forms of T(.x, nx)vII (x) on ¦£A (n = 3) ....................... 328
7.4.4 Implicit ABC for the Exterior Problem .............................. 333
7.5 Implicit ABCs for the Sound Wave Equation................................. 336
7.5.1 The Kirchhoff Formula for the 3-D Sound Wave Equation .................................................................... 337 References ............................................................................................... 338
Chapter 8 Nonlinear Artificial Boundary Conditions ................................ 341
8.1 The Burgers Equation .................................................................... 342
8.1.1 Nonlinear ABCs for the Burgers Equation.......................... 343
8.1.2 The Equivalent Initial Boundary Value Problem on the Bounded Computational Domain Di ............................. 346
8.2 The Kardar-Parisi-Zhang Equation................................................348
8.2.1 Nonlinear ABC for the K-P-Z Equation (D = 1)................ 349
8.2.2 Nonlinear ABC for the K-P-Z Equation (D = 2)................ 350
8.2.3 Nonlinear ABC for the K-P-Z Equation (D = 3)................ 353
8.3 The Cubic Nonlinear Schr.dinger Equation...................................354
8.3.1 Nonlinear Boundary Conditions on the Artificial Boundaries ¦²0 and ¦².1 ......................................................... 355
8.3.2 The Equivalent Initial Boundary Value Problem on the Bounded Domain [¨C1, 0] ¡Á [0, T ].................................. 356
8.4 Operator Splitting Method for Constructing Approximate Nonlinear ABCs..............................................................................358
8.4.1 The Local Absorbing ABC for the Linear Schr.dinger Equation..........................................................359
8.4.2 Finite Difference Approximation on the Bounded Computational Domain.......................................................360 References ............................................................................................... 362
Chapter 9 Applications to Problems with Singularity ............................... 365
9.1 The Modified Helmholtz Equation with a Singularity ................... 366
9.1.1 ABC Near Singular Points .................................................. 367
9.1.2 An Iteration Method Based on the ABC ............................ 368
9.2 The Interface Problem with a Singularity ...................................... 373
9.2.1 A Discrete Boundary Condition on the Artificial Boundary ¦£R ........................................................................ 374
9.2.2 Finite Element Approximation............................................379
9.3 The LinearElastic Problem with aSingularity..............................380
9.3.1 Discrete Boundary Condition on the Artificial Boundary ¦£R ........................................................................ 382
9.3.2 An Iteration Method Based on the ABC ............................ 390
9.4 The Stokes Equations with a Singularity ....................................... 393
9.4.1 The Discrete Boundary Condition on the Artificial Boundary ¦£R......................................................... 394
9.4.2 Singular Finite Element Approximation.............................. 403 References ............................................................................................... 406
Bibliography.................................................................................................. 409
