人工边界方法 英文【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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Introduction1

References5

Chapter 1 Global ABCs for Second Order Elliptic Equations9

1.1 Exterior Problem of Second Order Elliptic Equations9

1.2 Global ABCs for the Exterior Problem of 2-D Poisson Equation13

1.2.1 Steklov-Poincaré Mapping for the Exterior Problem of Laplace Equation14

1.2.2 The Reduced Boundary Value Problem on Ω17

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 Equation26

1.3.1 Exact and Approximate ABCs on the Spherical Artificial Boundary ГR26

1.3.2 Equivalent and Approximate Boundary Value Problems on the Bounded Computational Domain Ω30

1.3.3 Finite Element Approximation of the Variational Problem (1.3.30)34

1.4 Exterior Problem of the Modified Helmholtz Equation37

1.4.1 Global Boundary Condition of the Exterior Problem for the 2-D Modified Helmholtz Equation37

1.4.2 The Reduced Boundary Value Problem on the Computational Domain Ω39

1.4.3 Finite Element Approximation of the Reduced Boundary Value Problem45

1.4.4 Global Boundary Condition of the Exterior Problem for the 3-D Modified Helmholtz Equation47

1.5 Global ABCs for the Exterior Problems of the Helmholtz Equation49

1.5.1 Dirichlet to Sommerfeld Mapping of the Exterior Problem of the 2-D Helmholtz Equation49

1.5.2 Dirichlet to Sommerfeld Mapping of the Exterior Problem of the 3-D Helmholtz Equation55

References58

Chapter 2 Global ABCs for the Navier System and Stokes System61

2.1 Navier System and Stokes System61

2.2 The Exterior Problem of the 2-D Navier System64

2.2.1 The Global Boundary Condition on the Artificial Boundary ГR65

2.2.2 The Reduced Problem on the Bounded Domain71

2.2.3 The Finite Element Approximation for the Reduced Problem (2.2.59)77

2.3 Exterior Problem of the 2-D Stokes System79

2.3.1 Highly Accurate Approximate Artificial Boundary Condition80

2.3.2 Finite Element Approximation on the Computational Domain Ωi for the Reduced Problem84

2.4 Vector Fields on the Spherical Surface91

2.5 Global ABCs for the Exterior Problem of 3-D Navier System96

2.5.1 Highly Accurate Approximate ABCs96

2.5.2 Finite Element Approximation of the Variational Problem on the Bounded Computational Domain Ωi100

References111

Chapter 3 Global ABCs for Heat and Schr？dinger Equations115

3.1 Heat Equations on Unbounded Domains115

3.2 1-D Heat Equations on Unbounded Domains117

3.2.1 Exact Boundary Conditions on the Artificial Boundary ∑0117

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 Equations131

3.3.1 Exact and Approximate Conditions on the Artificial Boundary ∑R132

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 Equations140

3.4.1 Exact and Approximate Conditions on the Artificial Boundary ∑R140

3.4.2 Stability Analysis for the Reduced Initial Boundary Value Problem147

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 Domains151

3.6 1-D Schr？dinger Equation on Unbounded Domains152

3.6.1 The Reduced Initial Value Problem and its Finite Difference Approximation153

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 Equation166

3.7.1 Exact and Approximate Boundary Conditions on the Artificial Boundary ∑R167

3.7.2 Stability Analysis of the Reduced Approximate Initial Boundary Value Problem172

3.8 The Global Boundary Condition for the Exterior Problem of the 3-D Linear Schr？dinger Equation175

3.8.1 Exact and Approximate Boundary Conditions on the Artificial Boundary ∑R176

3.8.2 Stability Analysis of the Reduced Approximate Initial Boundary Value Problem183

References187

Chapter 4 ABCs for Wave Equation,Klein-Gordon Equation,and Linear KdV Equations189

4.1 1-D Wave Equation189

4.1.1 Transparent Boundary Conditions on the Artificial Boundaries ∑1 and ∑0190

4.2 2-D Wave Equation192

4.2.1 Absorbing Boundary Conditions193

4.2.2 The Initial Boundary Value Problem on the Bounded Computational Domain Di200

4.3 3-D Wave Equation203

4.3.1 Absorbing Boundary Condition on the Artificial Boundary ∑R204

4.3.2 The Equivalent and Approximate Initial Boundary Value Problem on the Bounded Computational Domain Di208

4.4 1-D Klein-Gordon Equation209

4.4.1 Absorbing Boundary Conditions on the Artificial Boundary ∑1,∑0210

4.4.2 The Initial Boundary Value Problem on the Bounded Computational Domain Di212

4.5 2-and 3-D Klein-Gordon Equations214

4.5.1 Absorbing Boundary Conditions on the Artificial Boundary ∑Rf2-D case)215

4.5.2 Absorbing Boundary Conditions on the Artificial Boundary ∑R(3-D case)220

4.5.3 The Initial Boundary Value Problem on the Bounded Computational Domain D223

4.6 Linear KdV Equation224

4.6.1 Absorbing Boundary Condition on the Artificial Boundaries ∑a and ∑b225

4.6.2 The Equivalent Initial Boundary Value Problem on the Bounded Computational Domain227

4.7 Appendix:Three Integration Formulas228

References232

Chapter 5 Local Artificial Boundary Conditions233

5.1 Local Boundary Conditions for Exterior Problems of the 2-D Poisson Equation234

5.1.1 Local Boundary Condition on the Artificial Bboundary ГR234

5.1.2 Finite Element Approximation Using the Local Boundary Condition and its Error Estimate236

5.2 Local Boundary Conditions for the 3-D Poisson Equation241

5.2.1 The Local Boundary Condition on the Artificial Boundary ГR for Problem(Ⅰ)242

5.2.2 Local Boundary Conditions on the Artificial Boundary ГR for Problem(Ⅱ)250

5.3 Local ABCs for Wave Equations on Unbounded Domains254

References257

Chapter 6 Discrete Artificial Boundary Conditions259

6.1 Boundary Condition on a Polygon Boundary for the 2-D Poisson Equation—The Method of Lines260

6.1.1 Discrete Boundary Conditions on Polygonal Boundaries260

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—Infinite Difference Method270

6.2.1 2-D Viscous Incompressible Flow in a Channel270

6.2.2 Discrete ABCs272

6.3 Numerical Simulation of Infinite Elastic Foundation—Infinite Element Method278

6.3.1 The Steklov-Poincarè on an Artificial Boundary of Line Segments279

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 KZ289

6.4 Discrete Absorbing Boundary Condition for the 1-D Klein-Gordon Equation—Z transform method292

6.4.1 Z Transform292

6.4.2 Discrete Absorbing ABC294

6.4.3 Finite Difference Approximation for the 1-D Klein-Gordon Equation on the Bounded Domain296

References297

Chapter 7 Implicit Artificial Boundary Conditions299

7.1 Implicit Boundary Condition for the Exterior Problem of the 2-D Poisson Equation300

7.1.1 The Single and Double Layer Potential,and Their Derivative for the 2-D Laplace Equation300

7.1.2 The Derivation of the Implicit ABC for the Exterior Problem of the 2-D Poisson Equation305

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 Equation310

7.3 ABC for the Exterior Problem of the Helmholtz Equation316

7.3.1 The Normal Derivative on ГA for the Double Layer Potential of the Helmholtz Equation318

7.4 Implicit ABCs for the Exterior Problems of the Navier System321

7.4.1 Fundamental Solution,Stress Operator,Single and Double Layer Potentials321

7.4.2 New Forms of T(？x,nx)vII(x)on ГA(n=2)323

7.4.3 New Forms of T(？x,nx)vII(x)on ГA(n=3)328

7.4.4 Implicit ABC for the Exterior Problem333

7.5 Implicit ABCs for the Sound Wave Equation336

7.5.1 The Kirchhoff Formula for the 3-D Sound Wave Equation337

References338

Chapter 8 Nonlinear Artificial Boundary Conditions341

8.1 The Burgers Equation342

8.1.1 Nonlinear ABCs for the Burgers Equation343

8.1.2 The Equivalent Initial Boundary Value Problem on the Bounded Computational Domain Di346

8.2 The Kardar-Parisi-Zhang Equation348

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 Equation354

8.3.1 Nonlinear Boundary Conditions on the Artificial Boundaries ∑0 and ∑1355

8.3.2 The Equivalent Initial Boundary Value Problem on the Bounded Domain [-1,0]×[0,T]356

8.4 Operator Splitting Method for Constructing Approximate Nonlinear ABCs358

8.4.1 The Local Absorbing ABC for the Linear Schr？dinger Equation359

8.4.2 Finite Difference Approximation on the Bounded Computational Domain360

References362

Chapter 9 Applications to Problems with Singularity365

9.1 The Modified Helmholtz Equation with a Singularity366

9.1.1 ABC Near Singular Points367

9.1.2 An Iteration Method Based on the ABC368

9.2 The Interface Problem with a Singularity373

9.2.1 A Discrete Boundary Condition on the Artificial Boundary ГR374

9.2.2 Finite Element Approximation379

9.3 The Linear Elastic Problem with a Singularity380

9.3.1 Discrete Boundary Condition on the Artificial Boundary ГR382

9.3.2 An Iteration Method Based on the ABC390

9.4 The Stokes Equations with a Singularity393

9.4.1 The Discrete Boundary Condition on the Artificial Boundary ГR394

9.4.2 Singular Finite Element Approximation403

References406

Bibliography409