The Method of Fundamental Solutions: Theory and Applications, livre ebook

The fundamental solutions (FS) satisfy the governing equations in a solution domain S, and then the numerical solutions can be found from the exterior and the interior boundary conditions on S. The resource nodes of FS are chosen outside S, distinctly from the case of the boundary element method (BEM). This method is called the method of fundamental solutions (MFS), which originated from Kupradze in 1963. The Laplace and the Helmholtz equations are studied in detail, and biharmonic equations and the Cauchy-Navier equation of linear elastostatics are also discussed. Moreover, better choices of source nodes are explored. The simplicity of numerical algorithms and high accuracy of numerical solutions are two remarkable advantages of the MFS. However, the ill-conditioning of the MFS is notorious, and the condition number (Cond) grows exponentially via the number of the unknowns used. In this book, the numerical algorithms are introduced and their characteristics are addressed. The main efforts are made to establish the theoretical analysis in errors and stability. The strict analysis (as well as choices of source nodes) in this book has provided the solid theoretical basis of the MFS, to grant it to become an effective and competent numerical method for partial differential equations (PDE). Based on some of our works published as journal papers, this book presents essential and important elements of the MFS. It is intended for researchers, graduated students, university students, computational experts, mathematicians and engineers.

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