基本介紹
圖書信息,圖書簡介,圖書前言,圖書目錄,
圖書信息
作者：Houde Han（韓厚德），Xiaonan Wu（巫孝南）圖書詳細信息：
ISBN：9787302303909
定價：120元
印次：1-1
裝幀：精裝
印刷日期：2012-11-23
ISBN：9787302303909
定價：120元
印次：1-1
裝幀：精裝
印刷日期：2012-11-23
圖書簡介
內容簡介
人工邊界方法是求解無界區域上偏微分方程（組）數值解的一個重要和有效的方法。人工邊界方法的核心問題是在人工邊界上如何對已知的問題找出問題的解滿足的準確
（或者高精度近似）的邊界條件。藉助於人工邊界方法，我們可將無界區域上的問題簡化為有界區域上的問題進行數值計算。本書系統地介紹了人工邊界方法的計算格式及其理論基礎。本書可以作為科學與工程計算專業研究生課程的教材，亦可以作為科學與工程計算專業科學技術人
人工邊界方法是求解無界區域上偏微分方程（組）數值解的一個重要和有效的方法。人工邊界方法的核心問題是在人工邊界上如何對已知的問題找出問題的解滿足的準確
（或者高精度近似）的邊界條件。藉助於人工邊界方法，我們可將無界區域上的問題簡化為有界區域上的問題進行數值計算。本書系統地介紹了人工邊界方法的計算格式及其理論基礎。本書可以作為科學與工程計算專業研究生課程的教材，亦可以作為科學與工程計算專業科學技術人
圖書前言
The artificial boundary method is an effective numerical method for solving partial differential equations on unbounded domains by applying artificial boundary conditions (ABCs) on the boundaries of the reduced bounded domains. With more than 30 years development, the artificial boundary method has reached maturity in recent years. It has been applied to various problems in scientific and engineering computations, and theoretical issues such as the convergence and error estimates of the artificial boundary method have been solved gradually. Based on the research works by the authors over many years and the works by other researchers, we have collected the methods and theories of the artificial boundary method and have presented them in this book.
The partial contents of this book were taught in the fall, 2005 and the spring, 2007 in the Department of Mathematical Sciences of Tsinghua University and the Department of Mathematics of University of Science and Technology of China, respectively.
This book has nine chapters, as listed below.
Chapter 1: Global ABCs for the Exterior Problem of Second Order Elliptic Equations
Chapter 2: Global ABCs for the Navier System and Stokes System
Chapter 3: Global ABCs for Heat and Schr.dinger Equations
Chapter 4: Absorbing Boundary Conditions for Wave Equation, Klein-Gordon Equation, and Linear KdV Equation
Chapter 5: Local ABCs
Chapter 6: Discrete ABCs
Chapter 7: Implicit ABCs
Chapter 8: Nonlinear ABCs
Chapter 9: Applications to Problems with Singularity
We have striven for accuracy and elegance in writing the book. However, errors are inevitable. We would be most grateful to learn of any errors in the book for the revision of future printing.
This book has benefited from works of other researchers, including our co-authors: Long-An Ying, Weizhu Bao, Zhongyi Huang, Chunxiong Zheng, Zhizhong Sun, Jicheng Jin, Dongsheng Yin, and Zhenli Xu. Professor Hermann Brunner has read through all the chapters of this book, and made numerous suggestions for improving the manuscript. We wish to express our appreciation for his kind help.
Houde Han, Xiaonan Wu
The partial contents of this book were taught in the fall, 2005 and the spring, 2007 in the Department of Mathematical Sciences of Tsinghua University and the Department of Mathematics of University of Science and Technology of China, respectively.
This book has nine chapters, as listed below.
Chapter 1: Global ABCs for the Exterior Problem of Second Order Elliptic Equations
Chapter 2: Global ABCs for the Navier System and Stokes System
Chapter 3: Global ABCs for Heat and Schr.dinger Equations
Chapter 4: Absorbing Boundary Conditions for Wave Equation, Klein-Gordon Equation, and Linear KdV Equation
Chapter 5: Local ABCs
Chapter 6: Discrete ABCs
Chapter 7: Implicit ABCs
Chapter 8: Nonlinear ABCs
Chapter 9: Applications to Problems with Singularity
We have striven for accuracy and elegance in writing the book. However, errors are inevitable. We would be most grateful to learn of any errors in the book for the revision of future printing.
This book has benefited from works of other researchers, including our co-authors: Long-An Ying, Weizhu Bao, Zhongyi Huang, Chunxiong Zheng, Zhizhong Sun, Jicheng Jin, Dongsheng Yin, and Zhenli Xu. Professor Hermann Brunner has read through all the chapters of this book, and made numerous suggestions for improving the manuscript. We wish to express our appreciation for his kind help.
Houde Han, Xiaonan Wu
圖書目錄
Introduction......................................................................................................1
References ...................................................................................................5
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-Poincaré Mapping for the Exterior Problem of Laplace Equation ..............................................................14
1.2.2 The Reduced Boundary Value Problem on .i.......................17
1.2.1 Steklov-Poincaré 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 ΓR...........................................................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
1.3 Global ABCs for the Exterior Problems of 3-D Poisson Equation...............................................................................26
1.3.1 Exact and Approximate ABCs on the Spherical Artificial Boundary ΓR...........................................................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
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 ΓR...........................................................65
2.2.2 The Reduced Problem on the Bounded Domain...................71
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 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
2.3.2 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
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 Σ0 ........................................................................ 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
3.2.2 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 ΣR......................................................... 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 ΣR......................................................... 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
3.3.1 Exact and Approximate Conditions on the Artificial Boundary ΣR......................................................... 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 ΣR......................................................... 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 ΣR ............................................. 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 ΣR ............................................. 176
3.8.2 Stability Analysis of the Reduced Approximate Initial
Boundary Value Problem .................................................... 183
References ............................................................................................... 187
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 ΣR ............................................. 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 ΣR ............................................. 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-D Wave Equation......................................................................... 189
4.1.1 Transparent Boundary Conditions on the Artificial Boundaries Σ1 and Σ0 ........................................... 190
4.2 2-D Wave Equation......................................................................... 192
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 3-D Wave Equation......................................................................... 203
4.3.1 Absorbing Boundary Condition on the Artificial Boundary ΣR......................................................... 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
4.3.2 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 Σ1, Σ0................................................................... 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
4.4.2 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 ΣR (2-D case) ...................................................... 215
4.5.2 Absorbing Boundary Conditions on the Artificial Boundary ΣR (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
4.5.2 Absorbing Boundary Conditions on the Artificial Boundary ΣR (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 Σa and Σb........................................................... 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
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 ΓR ...................................................................... 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
5.1.1 Local Boundary Condition on the Artificial Bboundary ΓR ...................................................................... 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 ΓR for Problem (I)............................................... 242
5.2.2 Local Boundary Conditions on the Artificial Boundary ΓR for Problem (II) ............................................. 250
5.3 Local ABCs for Wave Equations on Unbounded Domains............. 254
References ............................................................................................... 257
5.2.2 Local Boundary Conditions on the Artificial Boundary ΓR 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—The 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—Infinite Difference Method........................................................................... 270
6.2.1 2-D Viscous Incompressible Flow in a Channel................... 270
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—Infinite 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—Infinite Element Method.............................................................................. 278
6.3.1 The Steklov-Poincarè 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—Z transform method .............................. 292
6.4.1 Z Transform......................................................................... 292
6.3.1 The Steklov-Poincarè 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—Z 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
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
7.1.1 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 ΓA 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 ΓA (n = 2) ....................... 323
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 ΓA (n = 2) ....................... 323
7.4.3 New Forms of T(.x, nx)vII (x) on ΓA (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
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 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 Σ0 and Σ.1 ......................................................... 355
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 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
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 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 ΓR ........................................................................ 374
9.2.2 Finite Element Approximation............................................379
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 ΓR ........................................................................ 382
9.3.2 An Iteration Method Based on the ABC ............................ 390
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 ΓR......................................................... 394
9.4.2 Singular Finite Element Approximation.............................. 403
References ............................................................................................... 406
9.4.2 Singular Finite Element Approximation.............................. 403
References ............................................................................................... 406
Bibliography.................................................................................................. 409
