Preface..................................................... XI
Acknowledgements............................................ XV
CHAPTER 1
Introduction................................................. 1
1.1Historic Review.......................................... 1
1.2 BasicAlgorithms ........................................ 3
1.3Numerical Experiments .................................... 5
1.4Characteristics of the MFS ................................. 11
Part I.Laplace’s Equation ...................................... 15
CHAPTER 2
DirichletProblems ............................................ 19
2.1 BasicAlgorithms of MFS .................................. 19
2.2Preliminary Lemmas ...................................... 21
2.3 MainTheorems.......................................... 27
2.4Stability Analysis for Disk Domains .......................... 32
2.5 ProofMethodology ....................................... 39
CHAPTER 3
Neumann Problems ........................................... 41
3.1Introduction ............................................ 41
3.2 Method of Fundamental Solutions............................ 42
3.2.1Description of Algorithms ............................ 42
3.2.2 MainResults of Analysis and Their Applications ........... 44
3.3Stability Analysis of Disk Domains ........................... 45
3.4Stability Analysis for Bounded Simply-Connected Domains......... 49
3.4.1Trefftz Methods .................................... 50
3.4.2Collocation Trefftz Methods ........................... 52
3.5 ErrorEstimates ......................................... 54
3.6Concluding Remarks ...................................... 58
CHAPTER 4
OtherBoundary Problems ...................................... 61
4.1 MixedBoundary Condition Problems ......................... 61
4.2Interior Boundary Conditions ............................... 66
4.3 AnnularDomains ........................................ 70
CHAPTER 5
CombinedMethods............................................ 77
5.1Combined Methods ....................................... 77
5.2 VariantCombinations of FS and PS .......................... 79
5.2.1Simplified Hybrid Combination ........................ 79
5.2.2Hybrid Plus Penalty Combination ...................... 81
5.2.3Indirect Combination ................................ 84
5.3Combinations of MFS with Other Domain Methods .............. 86
5.3.1Combined with FEM ................................ 86
5.3.2Combined with FDM ................................ 87
5.3.3Combined with Radial Basis Functions................... 90
5.4Singularity Problems by Combination of MFS and MPS ........... 91
CHAPTER 6
SourceNodes on Elliptic Pseudo-Boundaries......................... 99
6.1Introduction ............................................ 99
6.2Algorithms of MFS ....................................... 101
6.3 ErrorAnalysis .......................................... 103
6.3.1Preliminary Lemmas ................................ 103
6.3.2 ErrorBounds ...................................... 107
6.4Stability Analysis ........................................ 113
6.5Selection of Pseudo-Boundaries .............................. 119
6.6Numerical Experiments .................................... 121
6.7Concluding Remarks ...................................... 124
Part II.Helmholtz’s Equations and Other Equations ................. 125
CHAPTER 7
HelmholtzEquations in Simply-Connected Domains ................... 127
7.1Introduction ............................................ 127
7.2Algorithms ............................................. 128
7.3 ErrorAnalysis for Bessel Functions ........................... 131
7.3.1Preliminary Lemmas ................................ 131
7.3.2 ErrorBounds with Small k ............................ 134
7.3.3Exploration of Bounded k ............................ 140
7.4Stability Analysis for Disk Domains .......................... 146
7.5Application to BKM ...................................... 149
CHAPTER 8
ExteriorProblems of Helmholtz Equation ........................... 155
8.1Introduction ............................................ 155
8.2Standard MFS .......................................... 157
8.2.1 BasicAlgorithms ................................... 157
8.2.2 BriefError Analysis ................................. 159
8.3Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161
8.4Modified MFS ........................................... 165
8.5 ErrorAnalysis for Modified MFS ............................ 166
8.5.1Preliminary Lemmas ................................ 167
8.5.2 ErrorBounds ...................................... 175
8.6Stability Analysis for Modified MFS .......................... 179
8.7Numerical Experiments .................................... 181
8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181
8.7.2Non-Circular Pseudo-Boundaries by Modified MFS ......... 186
8.8Concluding Remarks ...................................... 188
CHAPTER 9
HelmholtzEquations in Bounded Multiply-Connected Domains .......... 191
9.1Introduction ............................................ 191
9.2 BoundedSimply-Connected Domains ......................... 192
9.2.1Algorithms........................................ 192
9.2.2 BriefError Analysis ................................. 193
9.3 BoundedMultiply-Connected Domains ........................ 197
9.3.1Algorithms........................................ 197
9.3.2 ErrorAnalysis ..................................... 198
9.4Stability Analysis for Ring Domains .......................... 201
9.5Numerical Experiments .................................... 210
9.6Concluding Remarks ...................................... 214
CHAPTER 10
BiharmonicEquations ......................................... 215
10.1Introduction ........................................... 215
10.2Preliminary Lemmas ..................................... 217
10.3 ErrorBounds .......................................... 224
10.4Stability Analysis for Circular Domains ....................... 228
10.4.1Approaches for Seeking Eigenvalues.................... 228
10.4.2Eigenvalues λk(Φ) and λk(DΦ) ........................ 231
10.4.3Bounds of Condition Number ........................ 236
10.5Numerical Experiments ................................... 242
CHAPTER 11
ElasticProblems.............................................. 247
11.1Introduction ........................................... 247
11.2 LinearElastostatics Problems in 2D ......................... 247
11.2.1Basic Theory .................................... 247
11.2.2Traction Boundary Conditions ....................... 249
11.2.3Fundamental Solutions ............................. 250
11.2.4Particular Solutions ............................... 251
11.3 HTM,MFS and MPS .................................... 252
11.3.1Algorithms of HTM ............................... 252
11.3.2Algorithms of MFS and MPS ........................ 252
11.4 ErrorsBetween FS and PS ................................ 254
11.4.1Preliminary Lemmas ............................... 254
11.4.2Polynomials Pn Approximated by xn
11.4.3Other Proof for Theorem 11.4.1 ...................... 258
11.4.4 ThePolynomials LPn Approximated by Principal FS ...... 261
11.5 ErrorBounds for MFS and HTM ........................... 264
11.5.1 TheMFS ....................................... 264
11.5.2 TheHTM Using FS ............................... 266
11.6Numerical Experiments ................................... 268
11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271
11.7.1Preliminary Lemmas ............................... 271
11.7.2Addition Theorems ................................ 277
CHAPTER 12
CauchyProblems ............................................. 281
12.1Introduction ........................................... 281
12.2Algorithms of Collocation Trefftz Methods .................... 281
12.3Characteristics ......................................... 284
12.3.1Existence and Uniqueness ........................... 284
12.3.2Ill-Posedness of Inverse Problems ..................... 287
12.4 Errorand Stability Analysis ............................... 290
12.4.1Error Analysis ................................... 290
12.4.2Stability Analysis ................................. 291
12.5Applications to Cauchy Data .............................. 295
12.5.1Errors on Cauchy Boundary ......................... 295
12.5.2Sensitivity of Solutions on Cauchy Data ................ 296
12.6Numerical Experiments and Concluding Remarks ............... 297
CHAPTER 13
3D Problems................................................ 301
13.1Introduction ........................................... 301
13.2 Methodof Particular Solutions ............................. 302
13.3 Methodof Fundamental Solutions ........................... 309
VIIIContents
13.3.1Algorithms ...................................... 309
13.3.2 Linkto MPS..................................... 310
13.4 ErrorAnalysis for MFS ................................... 313
13.4.1Preliminary Lemmas ............................... 314
13.4.2Error Bounds .................................... 321
13.5Numerical Experiments ................................... 324
13.5.1Collocation Equations on Γ .......................... 324
13.5.2 ByMFS ........................................ 325
13.5.3 ByMPS ........................................ 330
13.6Concluding Remarks ..................................... 331
13.7Appendix: 3D Problems of Helmholtz Equations ................ 332
13.7.1Interior Dirichlet Problems .......................... 332
13.7.2Exterior Dirichlet Problems ......................... 333
Part III.Selection of Source Nodes and Related Topics ................ 335
CHAPTER 14
Comparisonsof MFS and MPS ................................... 339
14.1Introduction ........................................... 339
14.2 TwoBasis Boundary Methods.............................. 340
14.2.1Method of Particular Solutions ....................... 340
14.2.2Method of Fundamental Solutions..................... 342
14.3 TheMFS-QR .......................................... 346
14.3.1Algorithms in Elliptic Coordinates .................... 346
14.3.2Characteristics of MFS-QR .......................... 349
14.4Numerical Experiments and Comparisons ..................... 354
14.4.1Highly Smooth Boundary Data ....................... 355
14.4.2Boundary Data with Strong Singularity ................ 356
14.4.3Better Pseudo-Boundaries........................... 358
14.5Concluding Remarks ..................................... 360
CHAPTER 15
StabilityAnalysis for Smooth Closed Pseudo-Boundaries ............... 361
15.1Introduction ........................................... 361
15.2Relations Between FS and PS .............................. 362
15.3 Boundsof Cond for Non-Elliptic Pseudo-Boundaries ............. 365
CHAPTER 16
SingularityProblems from Source Functions; Removal Techniques ........ 375
16.1Introduction ........................................... 376
16.2Analytical Framework for CTM in [169] ...................... 378
16.3 ErrorBounds for Singular Solutions from (16.1.3) ............... 380
16.4Singularity for Polygonal Domains and Arbitrary Domains ........ 383
16.5Removal Techniques for Laplace’s Equation ................... 384
16.5.1 Forthe Case of Q∗ Outside Γ........................ 384
16.5.2 Forthe Case of Q∗ Inside Γ under theImage Node Existing . 386
16.6Numerical Experiments ................................... 388
16.7Applications to Amoeba-Like Domains ....................... 390
16.7.1Numerical Results................................. 390
16.7.2Removal Techniques Linked to Source Identification
Problems............................................ 394
16.8Concluding Remarks ..................................... 399
CHAPTER 17
SourceNodes on Pseudo Radial-Lines .............................. 401
17.1Introduction ........................................... 401
17.2 PseudoRadial-Lines ..................................... 404
17.2.1 OnePseudo Radial-Line ............................ 404
17.2.2 TwoPseudo Radial-Lines ........................... 408
17.3Stability Analysis ....................................... 409
17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409
17.3.2Upper Bound Estimates of Cond for Variant Case by Case II. 412
17.4Numerical Experiments ................................... 415
17.4.1 DiskDomains .................................... 415
17.4.2Non-Disk Domains ................................ 420
17.5Concluding Remarks ..................................... 424
Epilogue.................................................... 427
References.................................................. 431
Glossary ofSymbols ........................................... 443
Index ......................................................449
