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 Basic Algorithms ........................................ 3

1.3 Numerical Experiments .................................... 5

1.4 Characteristics of the MFS ................................. 11

Part I.Laplace’s Equation ...................................... 15

CHAPTER 2

Dirichlet Problems ............................................ 19

2.1 Basic Algorithms of MFS .................................. 19

2.2 Preliminary Lemmas ...................................... 21

2.3 MainTheorems.......................................... 27

2.4 Stability Analysis for Disk Domains .......................... 32

2.5 Proof Methodology ....................................... 39

CHAPTER 3

Neumann Problems ........................................... 41

3.1Introduction ............................................ 41

3.2 Method of Fundamental Solutions............................ 42

3.2.1 Description of Algorithms ............................ 42

3.2.2 Main Results of Analysis and Their Applications ........... 44

3.3 Stability Analysis of Disk Domains ........................... 45

3.4 Stability Analysis for Bounded Simply-Connected Domains......... 49

3.4.1Trefftz Methods .................................... 50

3.4.2 Collocation Trefftz Methods ........................... 52

3.5 Error Estimates ......................................... 54

3.6 Concluding Remarks ...................................... 58

CHAPTER 4

Other Boundary Problems ...................................... 61

4.1 Mixed Boundary Condition Problems ......................... 61

4.2 Interior Boundary Conditions ............................... 66

4.3 Annular Domains ........................................ 70

CHAPTER 5

Combined Methods............................................ 77

5.1 Combined Methods ....................................... 77

5.2 Variant Combinations of FS and PS .......................... 79

5.2.1Simplified Hybrid Combination ........................ 79

5.2.2 Hybrid Plus Penalty Combination ...................... 81

5.2.3 Indirect Combination ................................ 84

5.3 Combinations of MFS with Other Domain Methods .............. 86

5.3.1 Combined with FEM ................................ 86

5.3.2 Combined with FDM ................................ 87

5.3.3 Combined with Radial Basis Functions................... 90

5.4 Singularity Problems by Combination of MFS and MPS ........... 91

CHAPTER 6

Source Nodes on Elliptic Pseudo-Boundaries......................... 99

6.1 Introduction ............................................ 99

6.2 Algorithms of MFS ....................................... 101

6.3 Error Analysis .......................................... 103

6.3.1 Preliminary Lemmas ................................ 103

6.3.2 Error Bounds ...................................... 107

6.4 Stability Analysis ........................................ 113

6.5 Selection of Pseudo-Boundaries .............................. 119

6.6 Numerical Experiments .................................... 121

6.7 Concluding Remarks ...................................... 124

Part II. Helmholtz’s Equations and Other Equations ................. 125

CHAPTER 7

Helmholtz Equations in Simply-Connected Domains ................... 127

7.1 Introduction ............................................ 127

7.2 Algorithms ............................................. 128

7.3 Error Analysis for Bessel Functions ........................... 131

7.3.1 Preliminary Lemmas ................................ 131

7.3.2 Error Bounds with Small k ............................ 134

7.3.3 Exploration of Bounded k ............................ 140

7.4 Stability Analysis for Disk Domains .......................... 146

7.5 Application to BKM ...................................... 149

CHAPTER 8

Exterior Problems of Helmholtz Equation ........................... 155

8.1 Introduction ............................................ 155

8.2 Standard MFS .......................................... 157

8.2.1 Basic Algorithms ................................... 157

8.2.2 Brief Error Analysis ................................. 159

8.3 Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161

8.4 Modified MFS ........................................... 165

8.5 Error Analysis for Modified MFS ............................ 166

8.5.1 Preliminary Lemmas ................................ 167

8.5.2 Error Bounds ...................................... 175

8.6 Stability Analysis for Modified MFS .......................... 179

8.7 Numerical Experiments .................................... 181

8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181

8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS ......... 186

8.8 Concluding Remarks ...................................... 188

CHAPTER 9

Helmholtz Equations in Bounded Multiply-Connected Domains .......... 191

9.1 Introduction ............................................ 191

9.2 Bounded Simply-Connected Domains ......................... 192

9.2.1Algorithms........................................ 192

9.2.2 Brief Error Analysis ................................. 193

9.3 Bounded Multiply-Connected Domains ........................ 197

9.3.1Algorithms........................................ 197

9.3.2 Error Analysis ..................................... 198

9.4 Stability Analysis for Ring Domains .......................... 201

9.5 Numerical Experiments .................................... 210

9.6 Concluding Remarks ...................................... 214

CHAPTER 10

Biharmonic Equations ......................................... 215

10.1Introduction ........................................... 215

10.2 Preliminary Lemmas ..................................... 217

10.3 Error Bounds .......................................... 224

10.4 Stability Analysis for Circular Domains ....................... 228

10.4.1Approaches for Seeking Eigenvalues.................... 228

10.4.2 Eigenvalues λk(Φ) and λk(DΦ) ........................ 231

10.4.3 Bounds of Condition Number ........................ 236

10.5 Numerical Experiments ................................... 242

CHAPTER 11

Elastic Problems.............................................. 247

11.1Introduction ........................................... 247

11.2 Linear Elastostatics Problems in 2D ......................... 247

11.2.1Basic Theory .................................... 247

11.2.2 Traction Boundary Conditions ....................... 249

11.2.3 Fundamental Solutions ............................. 250

11.2.4 Particular Solutions ............................... 251

11.3 HTM,MFS and MPS .................................... 252

11.3.1 Algorithms of HTM ............................... 252

11.3.2 Algorithms of MFS and MPS ........................ 252

11.4  Errors Between FS and PS ................................ 254

11.4.1 Preliminary Lemmas ............................... 254

11.4.2 Polynomials Pn Approximated by xn

11.4.3 Other Proof for Theorem 11.4.1 ...................... 258

11.4.4 The Polynomials LPn Approximated by Principal FS ...... 261

11.5 Error Bounds for MFS and HTM ........................... 264

11.5.1 The MFS ....................................... 264

11.5.2 The HTM Using FS ............................... 266

11.6 Numerical Experiments ................................... 268

11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271

11.7.1 Preliminary Lemmas ............................... 271

11.7.2 Addition Theorems ................................ 277

CHAPTER 12

Cauchy Problems ............................................. 281

12.1Introduction ........................................... 281

12.2 Algorithms of Collocation Trefftz Methods .................... 281

12.3 Characteristics ......................................... 284

12.3.1 Existence and Uniqueness ........................... 284

12.3.2 Ill-Posedness of Inverse Problems ..................... 287

12.4 Errorand Stability Analysis ............................... 290

12.4.1Error Analysis ................................... 290

12.4.2 Stability Analysis ................................. 291

12.5 Applications to Cauchy Data .............................. 295

12.5.1 Errors on Cauchy Boundary ......................... 295

12.5.2 Sensitivity of Solutions on Cauchy Data ................ 296

12.6 Numerical Experiments and Concluding Remarks ............... 297

CHAPTER 13

3D Problems................................................ 301

13.1 Introduction ........................................... 301

13.2 Method of Particular Solutions ............................. 302

13.3 Method of Fundamental Solutions ........................... 309

13.3.1Algorithms ...................................... 309

13.3.2 Linkto MPS..................................... 310

13.4 Error Analysis for MFS ................................... 313

13.4.1 Preliminary Lemmas ............................... 314

13.4.2 Error Bounds .................................... 321

13.5 Numerical Experiments ................................... 324

13.5.1 Collocation Equations on Γ .......................... 324

13.5.2 By MFS ........................................ 325

13.5.3 By MPS ........................................ 330

13.6 Concluding Remarks ..................................... 331

13.7 Appendix: 3D Problems of Helmholtz Equations ................ 332

13.7.1 Interior Dirichlet Problems .......................... 332

13.7.2 Exterior Dirichlet Problems ......................... 333

Part III.Selection of Source Nodes and Related Topics ................ 335

CHAPTER 14

Comparisons of MFS and MPS ................................... 339

14.1Introduction ........................................... 339

14.2 Two Basis Boundary Methods.............................. 340

14.2.1 Method of Particular Solutions ....................... 340

14.2.2 Method of Fundamental Solutions..................... 342

14.3 The MFS-QR .......................................... 346

14.3.1Algorithms in Elliptic Coordinates .................... 346

14.3.2 Characteristics of MFS-QR .......................... 349

14.4 Numerical Experiments and Comparisons ..................... 354

14.4.1Highly Smooth Boundary Data ....................... 355

14.4.2 Boundary Data with Strong Singularity ................ 356

14.4.3 Better Pseudo-Boundaries........................... 358

14.5 Concluding Remarks ..................................... 360

CHAPTER 15

Stability Analysis for Smooth Closed Pseudo-Boundaries ............... 361

15.1Introduction ........................................... 361

15.2 Relations Between FS and PS .............................. 362

15.3 Bounds of Cond for Non-Elliptic Pseudo-Boundaries ............. 365

CHAPTER 16

Singularity Problems from Source Functions; Removal Techniques ........ 375

16.1Introduction ........................................... 376

16.2 Analytical Framework for CTM in [169] ...................... 378

16.3 Error Bounds for Singular Solutions from (16.1.3) ............... 380

16.4 Singularity for Polygonal Domains and Arbitrary Domains ........ 383

16.5 Removal Techniques for Laplace’s Equation ................... 384

16.5.1 For the Case of Q∗ Outside Γ........................ 384

16.5.2 For the Case of Q∗ Inside Γ under theImage Node Existing . 386

16.6 Numerical Experiments ................................... 388

16.7 Applications to Amoeba-Like Domains ....................... 390

16.7.1 Numerical Results................................. 390

16.7.2 Removal Techniques Linked to Source Identification

Problems............................................ 394

16.8 Concluding Remarks ..................................... 399

CHAPTER 17

Source Nodes on Pseudo Radial-Lines .............................. 401

17.1Introduction ........................................... 401

17.2 Pseudo Radial-Lines ..................................... 404

17.2.1 One Pseudo Radial-Line ............................ 404

17.2.2 Two Pseudo Radial-Lines ........................... 408

17.3 Stability Analysis ....................................... 409

17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409

17.3.2 Upper Bound Estimates of Cond for Variant Case by Case II. 412

17.4 Numerical Experiments ................................... 415

17.4.1 Disk Domains .................................... 415

17.4.2 Non-Disk Domains ................................ 420

17.5 Concluding Remarks ..................................... 424

Epilogue.................................................... 427

References.................................................. 431

Glossary of Symbols ........................................... 443

Index ......................................................449