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laboutique.edpsciences.fr-R002513 03 01 01 R002513 00 ED E107 00 01 01 The Method of Fundamental Solutions: Theory and Applications 01 eng 08 472 03 22 5450882 17 01 P15 EDP Sciences & Science Press 01 01 P15 EDP Sciences & Science Press 11 00 20231204 02 01 002629 03 9782759831722 15 9782759831722 03 01 D1 EDP Sciences 45 03 laboutique.edpsciences.fr-002629 03 01 01 002629 03 9782759831722 15 9782759831722 10 EA E107 10 00 01 R002513 ED E107 1 10 01 02 Current Natural Sciences 01 01 The Method of Fundamental Solutions: Theory and Applications 1 A01 01 A2219 Zi-Cai LI LI, Zi-Cai Zi-Cai LI Zi-Cai LI is Professor Emeritus of National Sun Yat-sen University. 2 A01 01 A2220 Hung-Tsai HUANG HUANG, Hung-Tsai Hung-Tsai HUANG Hung-Tsai HUANG is Professor of I-Shou University. 3 A01 01 A1982 Yimin WEI WEI, Yimin Yimin WEI Yimin WEI is a Professor at the School of Mathematical Sciences, Fudan University, who works on the numerical linear algebra and multilinear algebra. He is the author of more than 150 papers in the international journals and six monographs published by Science Press, Elsevier, Springer and World Scientific., etc. 4 A01 01 A2222 Liping ZHANG ZHANG, Liping Liping ZHANG Liping ZHANG is Associate Professor of Qufu Normal University. 01 eng 08 470 03 02 24 Izibook:Subject Mathématiques 24 Izibook:SubjectAndCategoryAndTags |Mathématiques| 20 MFS;PDEs;BEM;Solution Domain S;Laplace Equation;Helmholtz Equation;Biharmonic Equations;Condition Number;Errors and Stability;Source Nodes 10 2011 MAT019000 01 510 05 06 03 00 <blockquote>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.</blockquote> 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. 02 00 This book discusses the Method of Fundamental Solutions (MFS), a numerical technique for solving partial differential equations (PDEs). 04 00 Preface..................................................... XI<o:p></o:p>Acknowledgements............................................ XV<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction................................................. 1<o:p></o:p>1.1Historic Review.......................................... 1<o:p></o:p>1.2 Basic Algorithms ........................................ 3<o:p></o:p>1.3 Numerical Experiments .................................... 5<o:p></o:p>1.4 Characteristics of the MFS ................................. 11<o:p></o:p>Part I.Laplace’s Equation ...................................... 15<o:p></o:p>CHAPTER 2<o:p></o:p>Dirichlet Problems ............................................ 19<o:p></o:p>2.1 Basic Algorithms of MFS .................................. 19<o:p></o:p>2.2 Preliminary Lemmas ...................................... 21<o:p></o:p>2.3 MainTheorems.......................................... 27<o:p></o:p>2.4 Stability Analysis for Disk Domains .......................... 32<o:p></o:p>2.5 Proof Methodology ....................................... 39<o:p></o:p>CHAPTER 3<o:p></o:p>Neumann Problems ........................................... 41<o:p></o:p>3.1Introduction ............................................ 41<o:p></o:p>3.2 Method of Fundamental Solutions............................ 42<o:p></o:p>3.2.1 Description of Algorithms ............................ 42<o:p></o:p>3.2.2 Main Results of Analysis and Their Applications ........... 44<o:p></o:p>3.3 Stability Analysis of Disk Domains ........................... 45<o:p></o:p>3.4 Stability Analysis for Bounded Simply-Connected Domains......... 49<o:p></o:p>3.4.1Trefftz Methods .................................... 50<o:p></o:p>3.4.2 Collocation Trefftz Methods ........................... 52<o:p></o:p>3.5 Error Estimates ......................................... 54<o:p></o:p>3.6 Concluding Remarks ...................................... 58<o:p></o:p>CHAPTER 4<o:p></o:p>Other Boundary Problems ...................................... 61<o:p></o:p>4.1 Mixed Boundary Condition Problems ......................... 61<o:p></o:p>4.2 Interior Boundary Conditions ............................... 66<o:p></o:p>4.3 Annular Domains ........................................ 70<o:p></o:p>CHAPTER 5<o:p></o:p>Combined Methods............................................ 77<o:p></o:p>5.1 Combined Methods ....................................... 77<o:p></o:p>5.2 Variant Combinations of FS and PS .......................... 79<o:p></o:p>5.2.1Simplified Hybrid Combination ........................ 79<o:p></o:p>5.2.2 Hybrid Plus Penalty Combination ...................... 81<o:p></o:p>5.2.3 Indirect Combination ................................ 84<o:p></o:p>5.3 Combinations of MFS with Other Domain Methods .............. 86<o:p></o:p>5.3.1 Combined with FEM ................................ 86<o:p></o:p>5.3.2 Combined with FDM ................................ 87<o:p></o:p>5.3.3 Combined with Radial Basis Functions................... 90<o:p></o:p>5.4 Singularity Problems by Combination of MFS and MPS ........... 91<o:p></o:p>CHAPTER 6<o:p></o:p>Source Nodes on Elliptic Pseudo-Boundaries......................... 99<o:p></o:p>6.1 Introduction ............................................ 99<o:p></o:p>6.2 Algorithms of MFS ....................................... 101<o:p></o:p>6.3 Error Analysis .......................................... 103<o:p></o:p>6.3.1 Preliminary Lemmas ................................ 103<o:p></o:p>6.3.2 Error Bounds ...................................... 107<o:p></o:p>6.4 Stability Analysis ........................................ 113<o:p></o:p>6.5 Selection of Pseudo-Boundaries .............................. 119<o:p></o:p>6.6 Numerical Experiments .................................... 121<o:p></o:p>6.7 Concluding Remarks ...................................... 124<o:p></o:p>Part II. Helmholtz’s Equations and Other Equations ................. 125<o:p></o:p>CHAPTER 7<o:p></o:p>Helmholtz Equations in Simply-Connected Domains ................... 127<o:p></o:p>7.1 Introduction ............................................ 127<o:p></o:p>7.2 Algorithms ............................................. 128<o:p></o:p>7.3 Error Analysis for Bessel Functions ........................... 131<o:p></o:p>7.3.1 Preliminary Lemmas ................................ 131<o:p></o:p>7.3.2 Error Bounds with Small k ............................ 134<o:p></o:p>7.3.3 Exploration of Bounded k ............................ 140<o:p></o:p>7.4 Stability Analysis for Disk Domains .......................... 146<o:p></o:p>7.5 Application to BKM ...................................... 149<o:p></o:p>CHAPTER 8<o:p></o:p>Exterior Problems of Helmholtz Equation ........................... 155<o:p></o:p>8.1 Introduction ............................................ 155<o:p></o:p>8.2 Standard MFS .......................................... 157<o:p></o:p>8.2.1 Basic Algorithms ................................... 157<o:p></o:p>8.2.2 Brief Error Analysis ................................. 159<o:p></o:p>8.3 Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161<o:p></o:p>8.4 Modified MFS ........................................... 165<o:p></o:p>8.5 Error Analysis for Modified MFS ............................ 166<o:p></o:p>8.5.1 Preliminary Lemmas ................................ 167<o:p></o:p>8.5.2 Error Bounds ...................................... 175<o:p></o:p>8.6 Stability Analysis for Modified MFS .......................... 179<o:p></o:p>8.7 Numerical Experiments .................................... 181<o:p></o:p>8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181<o:p></o:p>8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS ......... 186<o:p></o:p>8.8 Concluding Remarks ...................................... 188<o:p></o:p>CHAPTER 9<o:p></o:p>Helmholtz Equations in Bounded Multiply-Connected Domains .......... 191<o:p></o:p>9.1 Introduction ............................................ 191<o:p></o:p>9.2 Bounded Simply-Connected Domains ......................... 192<o:p></o:p>9.2.1Algorithms........................................ 192<o:p></o:p>9.2.2 Brief Error Analysis ................................. 193<o:p></o:p>9.3 Bounded Multiply-Connected Domains ........................ 197<o:p></o:p>9.3.1Algorithms........................................ 197<o:p></o:p>9.3.2 Error Analysis ..................................... 198<o:p></o:p>9.4 Stability Analysis for Ring Domains .......................... 201<o:p></o:p>9.5 Numerical Experiments .................................... 210<o:p></o:p>9.6 Concluding Remarks ...................................... 214<o:p></o:p>CHAPTER 10<o:p></o:p>Biharmonic Equations ......................................... 215<o:p></o:p>10.1Introduction ........................................... 215<o:p></o:p>10.2 Preliminary Lemmas ..................................... 217<o:p></o:p>10.3 Error Bounds .......................................... 224<o:p></o:p>10.4 Stability Analysis for Circular Domains ....................... 228<o:p></o:p>10.4.1Approaches for Seeking Eigenvalues.................... 228<o:p></o:p>10.4.2 Eigenvalues λk(Φ) and λk(DΦ) ........................ 231<o:p></o:p>10.4.3 Bounds of Condition Number ........................ 236<o:p></o:p>10.5 Numerical Experiments ................................... 242<o:p></o:p>CHAPTER 11<o:p></o:p>Elastic Problems.............................................. 247<o:p></o:p>11.1Introduction ........................................... 247<o:p></o:p>11.2 Linear Elastostatics Problems in 2D ......................... 247<o:p></o:p>11.2.1Basic Theory .................................... 247<o:p></o:p>11.2.2 Traction Boundary Conditions ....................... 249<o:p></o:p>11.2.3 Fundamental Solutions ............................. 250<o:p></o:p>11.2.4 Particular Solutions ............................... 251<o:p></o:p>11.3 HTM,MFS and MPS .................................... 252<o:p></o:p>11.3.1 Algorithms of HTM ............................... 252<o:p></o:p>11.3.2 Algorithms of MFS and MPS ........................ 252<o:p></o:p>11.4  Errors Between FS and PS ................................ 254<o:p></o:p>11.4.1 Preliminary Lemmas ............................... 254<o:p></o:p>11.4.2 Polynomials Pn Approximated by xn<o:p></o:p>11.4.3 Other Proof for Theorem 11.4.1 ...................... 258<o:p></o:p>11.4.4 The Polynomials LPn Approximated by Principal FS ...... 261<o:p></o:p>11.5 Error Bounds for MFS and HTM ........................... 264<o:p></o:p>11.5.1 The MFS ....................................... 264<o:p></o:p>11.5.2 The HTM Using FS ............................... 266<o:p></o:p>11.6 Numerical Experiments ................................... 268<o:p></o:p>11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271<o:p></o:p>11.7.1 Preliminary Lemmas ............................... 271<o:p></o:p>11.7.2 Addition Theorems ................................ 277<o:p></o:p>CHAPTER 12<o:p></o:p>Cauchy Problems ............................................. 281<o:p></o:p>12.1Introduction ........................................... 281<o:p></o:p>12.2 Algorithms of Collocation Trefftz Methods .................... 281<o:p></o:p>12.3 Characteristics ......................................... 284<o:p></o:p>12.3.1 Existence and Uniqueness ........................... 284<o:p></o:p>12.3.2 Ill-Posedness of Inverse Problems ..................... 287<o:p></o:p>12.4 Errorand Stability Analysis ............................... 290<o:p></o:p>12.4.1Error Analysis ................................... 290<o:p></o:p>12.4.2 Stability Analysis ................................. 291<o:p></o:p>12.5 Applications to Cauchy Data .............................. 295<o:p></o:p>12.5.1 Errors on Cauchy Boundary ......................... 295<o:p></o:p>12.5.2 Sensitivity of Solutions on Cauchy Data ................ 296<o:p></o:p>12.6 Numerical Experiments and Concluding Remarks ............... 297<o:p></o:p>CHAPTER 13<o:p></o:p>3D Problems................................................ 301<o:p></o:p>13.1 Introduction ........................................... 301<o:p></o:p>13.2 Method of Particular Solutions ............................. 302<o:p></o:p>13.3 Method of Fundamental Solutions ........................... 30913.3.1Algorithms ...................................... 309<o:p></o:p>13.3.2 Linkto MPS..................................... 310<o:p></o:p>13.4 Error Analysis for MFS ................................... 313<o:p></o:p>13.4.1 Preliminary Lemmas ............................... 314<o:p></o:p>13.4.2 Error Bounds .................................... 321<o:p></o:p>13.5 Numerical Experiments ................................... 324<o:p></o:p>13.5.1 Collocation Equations on Γ .......................... 324<o:p></o:p>13.5.2 By MFS ........................................ 325<o:p></o:p>13.5.3 By MPS ........................................ 330<o:p></o:p>13.6 Concluding Remarks ..................................... 331<o:p></o:p>13.7 Appendix: 3D Problems of Helmholtz Equations ................ 332<o:p></o:p>13.7.1 Interior Dirichlet Problems .......................... 332<o:p></o:p>13.7.2 Exterior Dirichlet Problems ......................... 333<o:p></o:p>Part III.Selection of Source Nodes and Related Topics ................ 335<o:p></o:p>CHAPTER 14<o:p></o:p>Comparisons of MFS and MPS ................................... 339<o:p></o:p>14.1Introduction ........................................... 339<o:p></o:p>14.2 Two Basis Boundary Methods.............................. 340<o:p></o:p>14.2.1 Method of Particular Solutions ....................... 340<o:p></o:p>14.2.2 Method of Fundamental Solutions..................... 342<o:p></o:p>14.3 The MFS-QR .......................................... 346<o:p></o:p>14.3.1Algorithms in Elliptic Coordinates .................... 346<o:p></o:p>14.3.2 Characteristics of MFS-QR .......................... 349<o:p></o:p>14.4 Numerical Experiments and Comparisons ..................... 354<o:p></o:p>14.4.1Highly Smooth Boundary Data ....................... 355<o:p></o:p>14.4.2 Boundary Data with Strong Singularity ................ 356<o:p></o:p>14.4.3 Better Pseudo-Boundaries........................... 358<o:p></o:p>14.5 Concluding Remarks ..................................... 360<o:p></o:p>CHAPTER 15<o:p></o:p>Stability Analysis for Smooth Closed Pseudo-Boundaries ............... 361<o:p></o:p>15.1Introduction ........................................... 361<o:p></o:p>15.2 Relations Between FS and PS .............................. 362<o:p></o:p>15.3 Bounds of Cond for Non-Elliptic Pseudo-Boundaries ............. 365<o:p></o:p>CHAPTER 16<o:p></o:p>Singularity Problems from Source Functions; Removal Techniques ........ 375<o:p></o:p>16.1Introduction ........................................... 376<o:p></o:p>16.2 Analytical Framework for CTM in [169] ...................... 378<o:p></o:p>16.3 Error Bounds for Singular Solutions from (16.1.3) ............... 380<o:p></o:p>16.4 Singularity for Polygonal Domains and Arbitrary Domains ........ 383<o:p></o:p>16.5 Removal Techniques for Laplace’s Equation ................... 384<o:p></o:p>16.5.1 For the Case of Q∗ Outside Γ........................ 384<o:p></o:p>16.5.2 For the Case of Q∗ Inside Γ under theImage Node Existing . 386<o:p></o:p>16.6 Numerical Experiments ................................... 388<o:p></o:p>16.7 Applications to Amoeba-Like Domains ....................... 390<o:p></o:p>16.7.1 Numerical Results................................. 390<o:p></o:p>16.7.2 Removal Techniques Linked to Source Identification<o:p></o:p>Problems............................................ 394<o:p></o:p>16.8 Concluding Remarks ..................................... 399<o:p></o:p>CHAPTER 17<o:p></o:p>Source Nodes on Pseudo Radial-Lines .............................. 401<o:p></o:p>17.1Introduction ........................................... 401<o:p></o:p>17.2 Pseudo Radial-Lines ..................................... 404<o:p></o:p>17.2.1 One Pseudo Radial-Line ............................ 404<o:p></o:p>17.2.2 Two Pseudo Radial-Lines ........................... 408<o:p></o:p>17.3 Stability Analysis ....................................... 409<o:p></o:p>17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409<o:p></o:p>17.3.2 Upper Bound Estimates of Cond for Variant Case by Case II. 412<o:p></o:p>17.4 Numerical Experiments ................................... 415<o:p></o:p>17.4.1 Disk Domains .................................... 415<o:p></o:p>17.4.2 Non-Disk Domains ................................ 420<o:p></o:p>17.5 Concluding Remarks ..................................... 424<o:p></o:p>Epilogue.................................................... 427<o:p></o:p>References.................................................. 431<o:p></o:p>Glossary of Symbols ........................................... 443<o:p></o:p>Index ......................................................449<o:p></o:p> Preface..................................................... XI<o:p></o:p>Acknowledgements............................................ XV<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction................................................. 1<o:p></o:p>1.1Historic Review.......................................... 1<o:p></o:p>1.2 BasicAlgorithms ........................................ 3<o:p></o:p>1.3Numerical Experiments .................................... 5<o:p></o:p>1.4Characteristics of the MFS ................................. 11<o:p></o:p>Part I.Laplace’s Equation ...................................... 15<o:p></o:p>CHAPTER 2<o:p></o:p>DirichletProblems ............................................ 19<o:p></o:p>2.1 BasicAlgorithms of MFS .................................. 19<o:p></o:p>2.2Preliminary Lemmas ...................................... 21<o:p></o:p>2.3 MainTheorems.......................................... 27<o:p></o:p>2.4Stability Analysis for Disk Domains .......................... 32<o:p></o:p>2.5 ProofMethodology ....................................... 39<o:p></o:p>CHAPTER 3<o:p></o:p>Neumann Problems ........................................... 41<o:p></o:p>3.1Introduction ............................................ 41<o:p></o:p>3.2 Method of Fundamental Solutions............................ 42<o:p></o:p>3.2.1Description of Algorithms ............................ 42<o:p></o:p>3.2.2 MainResults of Analysis and Their Applications ........... 44<o:p></o:p>3.3Stability Analysis of Disk Domains ........................... 45<o:p></o:p>3.4Stability Analysis for Bounded Simply-Connected Domains......... 49<o:p></o:p>3.4.1Trefftz Methods .................................... 50<o:p></o:p>3.4.2Collocation Trefftz Methods ........................... 52<o:p></o:p>3.5 ErrorEstimates ......................................... 54<o:p></o:p>3.6Concluding Remarks ...................................... 58<o:p></o:p>CHAPTER 4<o:p></o:p>OtherBoundary Problems ...................................... 61<o:p></o:p>4.1 MixedBoundary Condition Problems ......................... 61<o:p></o:p>4.2Interior Boundary Conditions ............................... 66<o:p></o:p>4.3 AnnularDomains ........................................ 70<o:p></o:p>CHAPTER 5<o:p></o:p>CombinedMethods............................................ 77<o:p></o:p>5.1Combined Methods ....................................... 77<o:p></o:p>5.2 VariantCombinations of FS and PS .......................... 79<o:p></o:p>5.2.1Simplified Hybrid Combination ........................ 79<o:p></o:p>5.2.2Hybrid Plus Penalty Combination ...................... 81<o:p></o:p>5.2.3Indirect Combination ................................ 84<o:p></o:p>5.3Combinations of MFS with Other Domain Methods .............. 86<o:p></o:p>5.3.1Combined with FEM ................................ 86<o:p></o:p>5.3.2Combined with FDM ................................ 87<o:p></o:p>5.3.3Combined with Radial Basis Functions................... 90<o:p></o:p>5.4Singularity Problems by Combination of MFS and MPS ........... 91<o:p></o:p>CHAPTER 6<o:p></o:p>SourceNodes on Elliptic Pseudo-Boundaries......................... 99<o:p></o:p>6.1Introduction ............................................ 99<o:p></o:p>6.2Algorithms of MFS ....................................... 101<o:p></o:p>6.3 ErrorAnalysis .......................................... 103<o:p></o:p>6.3.1Preliminary Lemmas ................................ 103<o:p></o:p>6.3.2 ErrorBounds ...................................... 107<o:p></o:p>6.4Stability Analysis ........................................ 113<o:p></o:p>6.5Selection of Pseudo-Boundaries .............................. 119<o:p></o:p>6.6Numerical Experiments .................................... 121<o:p></o:p>6.7Concluding Remarks ...................................... 124<o:p></o:p>Part II.Helmholtz’s Equations and Other Equations ................. 125<o:p></o:p>CHAPTER 7<o:p></o:p>HelmholtzEquations in Simply-Connected Domains ................... 127<o:p></o:p>7.1Introduction ............................................ 127<o:p></o:p>7.2Algorithms ............................................. 128<o:p></o:p>7.3 ErrorAnalysis for Bessel Functions ........................... 131<o:p></o:p>7.3.1Preliminary Lemmas ................................ 131<o:p></o:p>7.3.2 ErrorBounds with Small k ............................ 134<o:p></o:p>7.3.3Exploration of Bounded k ............................ 140<o:p></o:p>7.4Stability Analysis for Disk Domains .......................... 146<o:p></o:p>7.5Application to BKM ...................................... 149<o:p></o:p>CHAPTER 8<o:p></o:p>ExteriorProblems of Helmholtz Equation ........................... 155<o:p></o:p>8.1Introduction ............................................ 155<o:p></o:p>8.2Standard MFS .......................................... 157<o:p></o:p>8.2.1 BasicAlgorithms ................................... 157<o:p></o:p>8.2.2 BriefError Analysis ................................. 159<o:p></o:p>8.3Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161<o:p></o:p>8.4Modified MFS ........................................... 165<o:p></o:p>8.5 ErrorAnalysis for Modified MFS ............................ 166<o:p></o:p>8.5.1Preliminary Lemmas ................................ 167<o:p></o:p>8.5.2 ErrorBounds ...................................... 175<o:p></o:p>8.6Stability Analysis for Modified MFS .......................... 179<o:p></o:p>8.7Numerical Experiments .................................... 181<o:p></o:p>8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181<o:p></o:p>8.7.2Non-Circular Pseudo-Boundaries by Modified MFS ......... 186<o:p></o:p>8.8Concluding Remarks ...................................... 188<o:p></o:p>CHAPTER 9<o:p></o:p>HelmholtzEquations in Bounded Multiply-Connected Domains .......... 191<o:p></o:p>9.1Introduction ............................................ 191<o:p></o:p>9.2 BoundedSimply-Connected Domains ......................... 192<o:p></o:p>9.2.1Algorithms........................................ 192<o:p></o:p>9.2.2 BriefError Analysis ................................. 193<o:p></o:p>9.3 BoundedMultiply-Connected Domains ........................ 197<o:p></o:p>9.3.1Algorithms........................................ 197<o:p></o:p>9.3.2 ErrorAnalysis ..................................... 198<o:p></o:p>9.4Stability Analysis for Ring Domains .......................... 201<o:p></o:p>9.5Numerical Experiments .................................... 210<o:p></o:p>9.6Concluding Remarks ...................................... 214<o:p></o:p>CHAPTER 10<o:p></o:p>BiharmonicEquations ......................................... 215<o:p></o:p>10.1Introduction ........................................... 215<o:p></o:p>10.2Preliminary Lemmas ..................................... 217<o:p></o:p>10.3 ErrorBounds .......................................... 224<o:p></o:p>10.4Stability Analysis for Circular Domains ....................... 228<o:p></o:p>10.4.1Approaches for Seeking Eigenvalues.................... 228<o:p></o:p>10.4.2Eigenvalues λk(Φ) and λk(DΦ) ........................ 231<o:p></o:p>10.4.3Bounds of Condition Number ........................ 236<o:p></o:p>10.5Numerical Experiments ................................... 242<o:p></o:p>CHAPTER 11<o:p></o:p>ElasticProblems.............................................. 247<o:p></o:p>11.1Introduction ........................................... 247<o:p></o:p>11.2 LinearElastostatics Problems in 2D ......................... 247<o:p></o:p>11.2.1Basic Theory .................................... 247<o:p></o:p>11.2.2Traction Boundary Conditions ....................... 249<o:p></o:p>11.2.3Fundamental Solutions ............................. 250<o:p></o:p>11.2.4Particular Solutions ............................... 251<o:p></o:p>11.3 HTM,MFS and MPS .................................... 252<o:p></o:p>11.3.1Algorithms of HTM ............................... 252<o:p></o:p>11.3.2Algorithms of MFS and MPS ........................ 252<o:p></o:p>11.4 ErrorsBetween FS and PS ................................ 254<o:p></o:p>11.4.1Preliminary Lemmas ............................... 254<o:p></o:p>11.4.2Polynomials Pn Approximated by xn<o:p></o:p>11.4.3Other Proof for Theorem 11.4.1 ...................... 258<o:p></o:p>11.4.4 ThePolynomials LPn Approximated by Principal FS ...... 261<o:p></o:p>11.5 ErrorBounds for MFS and HTM ........................... 264<o:p></o:p>11.5.1 TheMFS ....................................... 264<o:p></o:p>11.5.2 TheHTM Using FS ............................... 266<o:p></o:p>11.6Numerical Experiments ................................... 268<o:p></o:p>11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271<o:p></o:p>11.7.1Preliminary Lemmas ............................... 271<o:p></o:p>11.7.2Addition Theorems ................................ 277<o:p></o:p>CHAPTER 12<o:p></o:p>CauchyProblems ............................................. 281<o:p></o:p>12.1Introduction ........................................... 281<o:p></o:p>12.2Algorithms of Collocation Trefftz Methods .................... 281<o:p></o:p>12.3Characteristics ......................................... 284<o:p></o:p>12.3.1Existence and Uniqueness ........................... 284<o:p></o:p>12.3.2Ill-Posedness of Inverse Problems ..................... 287<o:p></o:p>12.4 Errorand Stability Analysis ............................... 290<o:p></o:p>12.4.1Error Analysis ................................... 290<o:p></o:p>12.4.2Stability Analysis ................................. 291<o:p></o:p>12.5Applications to Cauchy Data .............................. 295<o:p></o:p>12.5.1Errors on Cauchy Boundary ......................... 295<o:p></o:p>12.5.2Sensitivity of Solutions on Cauchy Data ................ 296<o:p></o:p>12.6Numerical Experiments and Concluding Remarks ............... 297<o:p></o:p>CHAPTER 13<o:p></o:p>3D Problems................................................ 301<o:p></o:p>13.1Introduction ........................................... 301<o:p></o:p>13.2 Methodof Particular Solutions ............................. 302<o:p></o:p>13.3 Methodof Fundamental Solutions ........................... 309<o:p></o:p>VIIIContents<o:p></o:p>13.3.1Algorithms ...................................... 309<o:p></o:p>13.3.2 Linkto MPS..................................... 310<o:p></o:p>13.4 ErrorAnalysis for MFS ................................... 313<o:p></o:p>13.4.1Preliminary Lemmas ............................... 314<o:p></o:p>13.4.2Error Bounds .................................... 321<o:p></o:p>13.5Numerical Experiments ................................... 324<o:p></o:p>13.5.1Collocation Equations on Γ .......................... 324<o:p></o:p>13.5.2 ByMFS ........................................ 325<o:p></o:p>13.5.3 ByMPS ........................................ 330<o:p></o:p>13.6Concluding Remarks ..................................... 331<o:p></o:p>13.7Appendix: 3D Problems of Helmholtz Equations ................ 332<o:p></o:p>13.7.1Interior Dirichlet Problems .......................... 332<o:p></o:p>13.7.2Exterior Dirichlet Problems ......................... 333<o:p></o:p>Part III.Selection of Source Nodes and Related Topics ................ 335<o:p></o:p>CHAPTER 14<o:p></o:p>Comparisonsof MFS and MPS ................................... 339<o:p></o:p>14.1Introduction ........................................... 339<o:p></o:p>14.2 TwoBasis Boundary Methods.............................. 340<o:p></o:p>14.2.1Method of Particular Solutions ....................... 340<o:p></o:p>14.2.2Method of Fundamental Solutions..................... 342<o:p></o:p>14.3 TheMFS-QR .......................................... 346<o:p></o:p>14.3.1Algorithms in Elliptic Coordinates .................... 346<o:p></o:p>14.3.2Characteristics of MFS-QR .......................... 349<o:p></o:p>14.4Numerical Experiments and Comparisons ..................... 354<o:p></o:p>14.4.1Highly Smooth Boundary Data ....................... 355<o:p></o:p>14.4.2Boundary Data with Strong Singularity ................ 356<o:p></o:p>14.4.3Better Pseudo-Boundaries........................... 358<o:p></o:p>14.5Concluding Remarks ..................................... 360<o:p></o:p>CHAPTER 15<o:p></o:p>StabilityAnalysis for Smooth Closed Pseudo-Boundaries ............... 361<o:p></o:p>15.1Introduction ........................................... 361<o:p></o:p>15.2Relations Between FS and PS .............................. 362<o:p></o:p>15.3 Boundsof Cond for Non-Elliptic Pseudo-Boundaries ............. 365<o:p></o:p>CHAPTER 16<o:p></o:p>SingularityProblems from Source Functions; Removal Techniques ........ 375<o:p></o:p>16.1Introduction ........................................... 376<o:p></o:p>16.2Analytical Framework for CTM in [169] ...................... 378<o:p></o:p>16.3 ErrorBounds for Singular Solutions from (16.1.3) ............... 380<o:p></o:p>16.4Singularity for Polygonal Domains and Arbitrary Domains ........ 383<o:p></o:p>16.5Removal Techniques for Laplace’s Equation ................... 384<o:p></o:p>16.5.1 Forthe Case of Q∗ Outside Γ........................ 384<o:p></o:p>16.5.2 Forthe Case of Q∗ Inside Γ under theImage Node Existing . 386<o:p></o:p>16.6Numerical Experiments ................................... 388<o:p></o:p>16.7Applications to Amoeba-Like Domains ....................... 390<o:p></o:p>16.7.1Numerical Results................................. 390<o:p></o:p>16.7.2Removal Techniques Linked to Source Identification<o:p></o:p>Problems............................................ 394<o:p></o:p>16.8Concluding Remarks ..................................... 399<o:p></o:p>CHAPTER 17<o:p></o:p>SourceNodes on Pseudo Radial-Lines .............................. 401<o:p></o:p>17.1Introduction ........................................... 401<o:p></o:p>17.2 PseudoRadial-Lines ..................................... 404<o:p></o:p>17.2.1 OnePseudo Radial-Line ............................ 404<o:p></o:p>17.2.2 TwoPseudo Radial-Lines ........................... 408<o:p></o:p>17.3Stability Analysis ....................................... 409<o:p></o:p>17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409<o:p></o:p>17.3.2Upper Bound Estimates of Cond for Variant Case by Case II. 412<o:p></o:p>17.4Numerical Experiments ................................... 415<o:p></o:p>17.4.1 DiskDomains .................................... 415<o:p></o:p>17.4.2Non-Disk Domains ................................ 420<o:p></o:p>17.5Concluding Remarks ..................................... 424<o:p></o:p>Epilogue.................................................... 427<o:p></o:p>References.................................................. 431<o:p></o:p>Glossary ofSymbols ........................................... 443<o:p></o:p>Index ......................................................449<o:p></o:p> 21 00 06 01 https://laboutique.edpsciences.fr/produit/1380/9782759831722/the-method-of-fundamental-solutions-theory-and-applications 01 00 03 02 https://laboutique.edpsciences.fr/system/product_pictures/data/009/983/053/original/9782759831715-The_Method_of_Fundamental_Solutions_couv-sofedis.jpg 17 14 20240913T175126+0200 01 P15 EDP Sciences & Science Press 01 01 P15 EDP Sciences & Science Press 04 11 00 20231204 01 00 20231204 19 00 20231204 2026 06 3052868830012 01 WORLD 13 03 9782759831715 15 9782759831715 06 03 9782759831739 15 9782759831739 WORLD 08 EDP Sciences & Science Press 04 01 00 20231204 03 01 D1 EDP Sciences 20 04 05 01 02 1 00 108.99 01 5.50 5.68 EUR 04 06 01 02 1 00 381.99 01 5.50 5.68 EUR 01 04 06 01 02 1 00 419.99 01 5.50 21.90 USD 01 laboutique.edpsciences.fr-002630 03 01 01 002630 03 9782759831739 15 9782759831739 10 EA 10 10 01 02 Current Natural Sciences 01 01 The Method of Fundamental Solutions: Theory and Applications 1 A01 01 A2219 Zi-Cai LI LI, Zi-Cai Zi-Cai LI Zi-Cai LI is Professor Emeritus of National Sun Yat-sen University. 2 A01 01 A2220 Hung-Tsai HUANG HUANG, Hung-Tsai Hung-Tsai HUANG Hung-Tsai HUANG is Professor of I-Shou University. 3 A01 01 A1982 Yimin WEI WEI, Yimin Yimin WEI Yimin WEI is a Professor at the School of Mathematical Sciences, Fudan University, who works on the numerical linear algebra and multilinear algebra. He is the author of more than 150 papers in the international journals and six monographs published by Science Press, Elsevier, Springer and World Scientific., etc. 4 A01 01 A2222 Liping ZHANG ZHANG, Liping Liping ZHANG Liping ZHANG is Associate Professor of Qufu Normal University. 01 eng 08 470 03 02 24 Izibook:Subject Mathématiques 24 Izibook:SubjectAndCategoryAndTags |Mathématiques| 20 MFS;PDEs;BEM;Solution Domain S;Laplace Equation;Helmholtz Equation;Biharmonic Equations;Condition Number;Errors and Stability;Source Nodes 10 2011 MAT019000 01 510 05 06 03 00 <blockquote>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.</blockquote> 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. 02 00 This book discusses the Method of Fundamental Solutions (MFS), a numerical technique for solving partial differential equations (PDEs). 04 00 Preface..................................................... XI<o:p></o:p>Acknowledgements............................................ XV<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction................................................. 1<o:p></o:p>1.1Historic Review.......................................... 1<o:p></o:p>1.2 Basic Algorithms ........................................ 3<o:p></o:p>1.3 Numerical Experiments .................................... 5<o:p></o:p>1.4 Characteristics of the MFS ................................. 11<o:p></o:p>Part I.Laplace’s Equation ...................................... 15<o:p></o:p>CHAPTER 2<o:p></o:p>Dirichlet Problems ............................................ 19<o:p></o:p>2.1 Basic Algorithms of MFS .................................. 19<o:p></o:p>2.2 Preliminary Lemmas ...................................... 21<o:p></o:p>2.3 MainTheorems.......................................... 27<o:p></o:p>2.4 Stability Analysis for Disk Domains .......................... 32<o:p></o:p>2.5 Proof Methodology ....................................... 39<o:p></o:p>CHAPTER 3<o:p></o:p>Neumann Problems ........................................... 41<o:p></o:p>3.1Introduction ............................................ 41<o:p></o:p>3.2 Method of Fundamental Solutions............................ 42<o:p></o:p>3.2.1 Description of Algorithms ............................ 42<o:p></o:p>3.2.2 Main Results of Analysis and Their Applications ........... 44<o:p></o:p>3.3 Stability Analysis of Disk Domains ........................... 45<o:p></o:p>3.4 Stability Analysis for Bounded Simply-Connected Domains......... 49<o:p></o:p>3.4.1Trefftz Methods .................................... 50<o:p></o:p>3.4.2 Collocation Trefftz Methods ........................... 52<o:p></o:p>3.5 Error Estimates ......................................... 54<o:p></o:p>3.6 Concluding Remarks ...................................... 58<o:p></o:p>CHAPTER 4<o:p></o:p>Other Boundary Problems ...................................... 61<o:p></o:p>4.1 Mixed Boundary Condition Problems ......................... 61<o:p></o:p>4.2 Interior Boundary Conditions ............................... 66<o:p></o:p>4.3 Annular Domains ........................................ 70<o:p></o:p>CHAPTER 5<o:p></o:p>Combined Methods............................................ 77<o:p></o:p>5.1 Combined Methods ....................................... 77<o:p></o:p>5.2 Variant Combinations of FS and PS .......................... 79<o:p></o:p>5.2.1Simplified Hybrid Combination ........................ 79<o:p></o:p>5.2.2 Hybrid Plus Penalty Combination ...................... 81<o:p></o:p>5.2.3 Indirect Combination ................................ 84<o:p></o:p>5.3 Combinations of MFS with Other Domain Methods .............. 86<o:p></o:p>5.3.1 Combined with FEM ................................ 86<o:p></o:p>5.3.2 Combined with FDM ................................ 87<o:p></o:p>5.3.3 Combined with Radial Basis Functions................... 90<o:p></o:p>5.4 Singularity Problems by Combination of MFS and MPS ........... 91<o:p></o:p>CHAPTER 6<o:p></o:p>Source Nodes on Elliptic Pseudo-Boundaries......................... 99<o:p></o:p>6.1 Introduction ............................................ 99<o:p></o:p>6.2 Algorithms of MFS ....................................... 101<o:p></o:p>6.3 Error Analysis .......................................... 103<o:p></o:p>6.3.1 Preliminary Lemmas ................................ 103<o:p></o:p>6.3.2 Error Bounds ...................................... 107<o:p></o:p>6.4 Stability Analysis ........................................ 113<o:p></o:p>6.5 Selection of Pseudo-Boundaries .............................. 119<o:p></o:p>6.6 Numerical Experiments .................................... 121<o:p></o:p>6.7 Concluding Remarks ...................................... 124<o:p></o:p>Part II. Helmholtz’s Equations and Other Equations ................. 125<o:p></o:p>CHAPTER 7<o:p></o:p>Helmholtz Equations in Simply-Connected Domains ................... 127<o:p></o:p>7.1 Introduction ............................................ 127<o:p></o:p>7.2 Algorithms ............................................. 128<o:p></o:p>7.3 Error Analysis for Bessel Functions ........................... 131<o:p></o:p>7.3.1 Preliminary Lemmas ................................ 131<o:p></o:p>7.3.2 Error Bounds with Small k ............................ 134<o:p></o:p>7.3.3 Exploration of Bounded k ............................ 140<o:p></o:p>7.4 Stability Analysis for Disk Domains .......................... 146<o:p></o:p>7.5 Application to BKM ...................................... 149<o:p></o:p>CHAPTER 8<o:p></o:p>Exterior Problems of Helmholtz Equation ........................... 155<o:p></o:p>8.1 Introduction ............................................ 155<o:p></o:p>8.2 Standard MFS .......................................... 157<o:p></o:p>8.2.1 Basic Algorithms ................................... 157<o:p></o:p>8.2.2 Brief Error Analysis ................................. 159<o:p></o:p>8.3 Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161<o:p></o:p>8.4 Modified MFS ........................................... 165<o:p></o:p>8.5 Error Analysis for Modified MFS ............................ 166<o:p></o:p>8.5.1 Preliminary Lemmas ................................ 167<o:p></o:p>8.5.2 Error Bounds ...................................... 175<o:p></o:p>8.6 Stability Analysis for Modified MFS .......................... 179<o:p></o:p>8.7 Numerical Experiments .................................... 181<o:p></o:p>8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181<o:p></o:p>8.7.2 Non-Circular Pseudo-Boundaries by Modified MFS ......... 186<o:p></o:p>8.8 Concluding Remarks ...................................... 188<o:p></o:p>CHAPTER 9<o:p></o:p>Helmholtz Equations in Bounded Multiply-Connected Domains .......... 191<o:p></o:p>9.1 Introduction ............................................ 191<o:p></o:p>9.2 Bounded Simply-Connected Domains ......................... 192<o:p></o:p>9.2.1Algorithms........................................ 192<o:p></o:p>9.2.2 Brief Error Analysis ................................. 193<o:p></o:p>9.3 Bounded Multiply-Connected Domains ........................ 197<o:p></o:p>9.3.1Algorithms........................................ 197<o:p></o:p>9.3.2 Error Analysis ..................................... 198<o:p></o:p>9.4 Stability Analysis for Ring Domains .......................... 201<o:p></o:p>9.5 Numerical Experiments .................................... 210<o:p></o:p>9.6 Concluding Remarks ...................................... 214<o:p></o:p>CHAPTER 10<o:p></o:p>Biharmonic Equations ......................................... 215<o:p></o:p>10.1Introduction ........................................... 215<o:p></o:p>10.2 Preliminary Lemmas ..................................... 217<o:p></o:p>10.3 Error Bounds .......................................... 224<o:p></o:p>10.4 Stability Analysis for Circular Domains ....................... 228<o:p></o:p>10.4.1Approaches for Seeking Eigenvalues.................... 228<o:p></o:p>10.4.2 Eigenvalues λk(Φ) and λk(DΦ) ........................ 231<o:p></o:p>10.4.3 Bounds of Condition Number ........................ 236<o:p></o:p>10.5 Numerical Experiments ................................... 242<o:p></o:p>CHAPTER 11<o:p></o:p>Elastic Problems.............................................. 247<o:p></o:p>11.1Introduction ........................................... 247<o:p></o:p>11.2 Linear Elastostatics Problems in 2D ......................... 247<o:p></o:p>11.2.1Basic Theory .................................... 247<o:p></o:p>11.2.2 Traction Boundary Conditions ....................... 249<o:p></o:p>11.2.3 Fundamental Solutions ............................. 250<o:p></o:p>11.2.4 Particular Solutions ............................... 251<o:p></o:p>11.3 HTM,MFS and MPS .................................... 252<o:p></o:p>11.3.1 Algorithms of HTM ............................... 252<o:p></o:p>11.3.2 Algorithms of MFS and MPS ........................ 252<o:p></o:p>11.4  Errors Between FS and PS ................................ 254<o:p></o:p>11.4.1 Preliminary Lemmas ............................... 254<o:p></o:p>11.4.2 Polynomials Pn Approximated by xn<o:p></o:p>11.4.3 Other Proof for Theorem 11.4.1 ...................... 258<o:p></o:p>11.4.4 The Polynomials LPn Approximated by Principal FS ...... 261<o:p></o:p>11.5 Error Bounds for MFS and HTM ........................... 264<o:p></o:p>11.5.1 The MFS ....................................... 264<o:p></o:p>11.5.2 The HTM Using FS ............................... 266<o:p></o:p>11.6 Numerical Experiments ................................... 268<o:p></o:p>11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271<o:p></o:p>11.7.1 Preliminary Lemmas ............................... 271<o:p></o:p>11.7.2 Addition Theorems ................................ 277<o:p></o:p>CHAPTER 12<o:p></o:p>Cauchy Problems ............................................. 281<o:p></o:p>12.1Introduction ........................................... 281<o:p></o:p>12.2 Algorithms of Collocation Trefftz Methods .................... 281<o:p></o:p>12.3 Characteristics ......................................... 284<o:p></o:p>12.3.1 Existence and Uniqueness ........................... 284<o:p></o:p>12.3.2 Ill-Posedness of Inverse Problems ..................... 287<o:p></o:p>12.4 Errorand Stability Analysis ............................... 290<o:p></o:p>12.4.1Error Analysis ................................... 290<o:p></o:p>12.4.2 Stability Analysis ................................. 291<o:p></o:p>12.5 Applications to Cauchy Data .............................. 295<o:p></o:p>12.5.1 Errors on Cauchy Boundary ......................... 295<o:p></o:p>12.5.2 Sensitivity of Solutions on Cauchy Data ................ 296<o:p></o:p>12.6 Numerical Experiments and Concluding Remarks ............... 297<o:p></o:p>CHAPTER 13<o:p></o:p>3D Problems................................................ 301<o:p></o:p>13.1 Introduction ........................................... 301<o:p></o:p>13.2 Method of Particular Solutions ............................. 302<o:p></o:p>13.3 Method of Fundamental Solutions ........................... 30913.3.1Algorithms ...................................... 309<o:p></o:p>13.3.2 Linkto MPS..................................... 310<o:p></o:p>13.4 Error Analysis for MFS ................................... 313<o:p></o:p>13.4.1 Preliminary Lemmas ............................... 314<o:p></o:p>13.4.2 Error Bounds .................................... 321<o:p></o:p>13.5 Numerical Experiments ................................... 324<o:p></o:p>13.5.1 Collocation Equations on Γ .......................... 324<o:p></o:p>13.5.2 By MFS ........................................ 325<o:p></o:p>13.5.3 By MPS ........................................ 330<o:p></o:p>13.6 Concluding Remarks ..................................... 331<o:p></o:p>13.7 Appendix: 3D Problems of Helmholtz Equations ................ 332<o:p></o:p>13.7.1 Interior Dirichlet Problems .......................... 332<o:p></o:p>13.7.2 Exterior Dirichlet Problems ......................... 333<o:p></o:p>Part III.Selection of Source Nodes and Related Topics ................ 335<o:p></o:p>CHAPTER 14<o:p></o:p>Comparisons of MFS and MPS ................................... 339<o:p></o:p>14.1Introduction ........................................... 339<o:p></o:p>14.2 Two Basis Boundary Methods.............................. 340<o:p></o:p>14.2.1 Method of Particular Solutions ....................... 340<o:p></o:p>14.2.2 Method of Fundamental Solutions..................... 342<o:p></o:p>14.3 The MFS-QR .......................................... 346<o:p></o:p>14.3.1Algorithms in Elliptic Coordinates .................... 346<o:p></o:p>14.3.2 Characteristics of MFS-QR .......................... 349<o:p></o:p>14.4 Numerical Experiments and Comparisons ..................... 354<o:p></o:p>14.4.1Highly Smooth Boundary Data ....................... 355<o:p></o:p>14.4.2 Boundary Data with Strong Singularity ................ 356<o:p></o:p>14.4.3 Better Pseudo-Boundaries........................... 358<o:p></o:p>14.5 Concluding Remarks ..................................... 360<o:p></o:p>CHAPTER 15<o:p></o:p>Stability Analysis for Smooth Closed Pseudo-Boundaries ............... 361<o:p></o:p>15.1Introduction ........................................... 361<o:p></o:p>15.2 Relations Between FS and PS .............................. 362<o:p></o:p>15.3 Bounds of Cond for Non-Elliptic Pseudo-Boundaries ............. 365<o:p></o:p>CHAPTER 16<o:p></o:p>Singularity Problems from Source Functions; Removal Techniques ........ 375<o:p></o:p>16.1Introduction ........................................... 376<o:p></o:p>16.2 Analytical Framework for CTM in [169] ...................... 378<o:p></o:p>16.3 Error Bounds for Singular Solutions from (16.1.3) ............... 380<o:p></o:p>16.4 Singularity for Polygonal Domains and Arbitrary Domains ........ 383<o:p></o:p>16.5 Removal Techniques for Laplace’s Equation ................... 384<o:p></o:p>16.5.1 For the Case of Q∗ Outside Γ........................ 384<o:p></o:p>16.5.2 For the Case of Q∗ Inside Γ under theImage Node Existing . 386<o:p></o:p>16.6 Numerical Experiments ................................... 388<o:p></o:p>16.7 Applications to Amoeba-Like Domains ....................... 390<o:p></o:p>16.7.1 Numerical Results................................. 390<o:p></o:p>16.7.2 Removal Techniques Linked to Source Identification<o:p></o:p>Problems............................................ 394<o:p></o:p>16.8 Concluding Remarks ..................................... 399<o:p></o:p>CHAPTER 17<o:p></o:p>Source Nodes on Pseudo Radial-Lines .............................. 401<o:p></o:p>17.1Introduction ........................................... 401<o:p></o:p>17.2 Pseudo Radial-Lines ..................................... 404<o:p></o:p>17.2.1 One Pseudo Radial-Line ............................ 404<o:p></o:p>17.2.2 Two Pseudo Radial-Lines ........................... 408<o:p></o:p>17.3 Stability Analysis ....................................... 409<o:p></o:p>17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409<o:p></o:p>17.3.2 Upper Bound Estimates of Cond for Variant Case by Case II. 412<o:p></o:p>17.4 Numerical Experiments ................................... 415<o:p></o:p>17.4.1 Disk Domains .................................... 415<o:p></o:p>17.4.2 Non-Disk Domains ................................ 420<o:p></o:p>17.5 Concluding Remarks ..................................... 424<o:p></o:p>Epilogue.................................................... 427<o:p></o:p>References.................................................. 431<o:p></o:p>Glossary of Symbols ........................................... 443<o:p></o:p>Index ......................................................449<o:p></o:p> Preface..................................................... XI<o:p></o:p>Acknowledgements............................................ XV<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction................................................. 1<o:p></o:p>1.1Historic Review.......................................... 1<o:p></o:p>1.2 BasicAlgorithms ........................................ 3<o:p></o:p>1.3Numerical Experiments .................................... 5<o:p></o:p>1.4Characteristics of the MFS ................................. 11<o:p></o:p>Part I.Laplace’s Equation ...................................... 15<o:p></o:p>CHAPTER 2<o:p></o:p>DirichletProblems ............................................ 19<o:p></o:p>2.1 BasicAlgorithms of MFS .................................. 19<o:p></o:p>2.2Preliminary Lemmas ...................................... 21<o:p></o:p>2.3 MainTheorems.......................................... 27<o:p></o:p>2.4Stability Analysis for Disk Domains .......................... 32<o:p></o:p>2.5 ProofMethodology ....................................... 39<o:p></o:p>CHAPTER 3<o:p></o:p>Neumann Problems ........................................... 41<o:p></o:p>3.1Introduction ............................................ 41<o:p></o:p>3.2 Method of Fundamental Solutions............................ 42<o:p></o:p>3.2.1Description of Algorithms ............................ 42<o:p></o:p>3.2.2 MainResults of Analysis and Their Applications ........... 44<o:p></o:p>3.3Stability Analysis of Disk Domains ........................... 45<o:p></o:p>3.4Stability Analysis for Bounded Simply-Connected Domains......... 49<o:p></o:p>3.4.1Trefftz Methods .................................... 50<o:p></o:p>3.4.2Collocation Trefftz Methods ........................... 52<o:p></o:p>3.5 ErrorEstimates ......................................... 54<o:p></o:p>3.6Concluding Remarks ...................................... 58<o:p></o:p>CHAPTER 4<o:p></o:p>OtherBoundary Problems ...................................... 61<o:p></o:p>4.1 MixedBoundary Condition Problems ......................... 61<o:p></o:p>4.2Interior Boundary Conditions ............................... 66<o:p></o:p>4.3 AnnularDomains ........................................ 70<o:p></o:p>CHAPTER 5<o:p></o:p>CombinedMethods............................................ 77<o:p></o:p>5.1Combined Methods ....................................... 77<o:p></o:p>5.2 VariantCombinations of FS and PS .......................... 79<o:p></o:p>5.2.1Simplified Hybrid Combination ........................ 79<o:p></o:p>5.2.2Hybrid Plus Penalty Combination ...................... 81<o:p></o:p>5.2.3Indirect Combination ................................ 84<o:p></o:p>5.3Combinations of MFS with Other Domain Methods .............. 86<o:p></o:p>5.3.1Combined with FEM ................................ 86<o:p></o:p>5.3.2Combined with FDM ................................ 87<o:p></o:p>5.3.3Combined with Radial Basis Functions................... 90<o:p></o:p>5.4Singularity Problems by Combination of MFS and MPS ........... 91<o:p></o:p>CHAPTER 6<o:p></o:p>SourceNodes on Elliptic Pseudo-Boundaries......................... 99<o:p></o:p>6.1Introduction ............................................ 99<o:p></o:p>6.2Algorithms of MFS ....................................... 101<o:p></o:p>6.3 ErrorAnalysis .......................................... 103<o:p></o:p>6.3.1Preliminary Lemmas ................................ 103<o:p></o:p>6.3.2 ErrorBounds ...................................... 107<o:p></o:p>6.4Stability Analysis ........................................ 113<o:p></o:p>6.5Selection of Pseudo-Boundaries .............................. 119<o:p></o:p>6.6Numerical Experiments .................................... 121<o:p></o:p>6.7Concluding Remarks ...................................... 124<o:p></o:p>Part II.Helmholtz’s Equations and Other Equations ................. 125<o:p></o:p>CHAPTER 7<o:p></o:p>HelmholtzEquations in Simply-Connected Domains ................... 127<o:p></o:p>7.1Introduction ............................................ 127<o:p></o:p>7.2Algorithms ............................................. 128<o:p></o:p>7.3 ErrorAnalysis for Bessel Functions ........................... 131<o:p></o:p>7.3.1Preliminary Lemmas ................................ 131<o:p></o:p>7.3.2 ErrorBounds with Small k ............................ 134<o:p></o:p>7.3.3Exploration of Bounded k ............................ 140<o:p></o:p>7.4Stability Analysis for Disk Domains .......................... 146<o:p></o:p>7.5Application to BKM ...................................... 149<o:p></o:p>CHAPTER 8<o:p></o:p>ExteriorProblems of Helmholtz Equation ........................... 155<o:p></o:p>8.1Introduction ............................................ 155<o:p></o:p>8.2Standard MFS .......................................... 157<o:p></o:p>8.2.1 BasicAlgorithms ................................... 157<o:p></o:p>8.2.2 BriefError Analysis ................................. 159<o:p></o:p>8.3Numerical Characteristics of Spurious Eigenvalues by MFS ......... 161<o:p></o:p>8.4Modified MFS ........................................... 165<o:p></o:p>8.5 ErrorAnalysis for Modified MFS ............................ 166<o:p></o:p>8.5.1Preliminary Lemmas ................................ 167<o:p></o:p>8.5.2 ErrorBounds ...................................... 175<o:p></o:p>8.6Stability Analysis for Modified MFS .......................... 179<o:p></o:p>8.7Numerical Experiments .................................... 181<o:p></o:p>8.7.1Circular Pseudo-Boundaries by Two MFS ................ 181<o:p></o:p>8.7.2Non-Circular Pseudo-Boundaries by Modified MFS ......... 186<o:p></o:p>8.8Concluding Remarks ...................................... 188<o:p></o:p>CHAPTER 9<o:p></o:p>HelmholtzEquations in Bounded Multiply-Connected Domains .......... 191<o:p></o:p>9.1Introduction ............................................ 191<o:p></o:p>9.2 BoundedSimply-Connected Domains ......................... 192<o:p></o:p>9.2.1Algorithms........................................ 192<o:p></o:p>9.2.2 BriefError Analysis ................................. 193<o:p></o:p>9.3 BoundedMultiply-Connected Domains ........................ 197<o:p></o:p>9.3.1Algorithms........................................ 197<o:p></o:p>9.3.2 ErrorAnalysis ..................................... 198<o:p></o:p>9.4Stability Analysis for Ring Domains .......................... 201<o:p></o:p>9.5Numerical Experiments .................................... 210<o:p></o:p>9.6Concluding Remarks ...................................... 214<o:p></o:p>CHAPTER 10<o:p></o:p>BiharmonicEquations ......................................... 215<o:p></o:p>10.1Introduction ........................................... 215<o:p></o:p>10.2Preliminary Lemmas ..................................... 217<o:p></o:p>10.3 ErrorBounds .......................................... 224<o:p></o:p>10.4Stability Analysis for Circular Domains ....................... 228<o:p></o:p>10.4.1Approaches for Seeking Eigenvalues.................... 228<o:p></o:p>10.4.2Eigenvalues λk(Φ) and λk(DΦ) ........................ 231<o:p></o:p>10.4.3Bounds of Condition Number ........................ 236<o:p></o:p>10.5Numerical Experiments ................................... 242<o:p></o:p>CHAPTER 11<o:p></o:p>ElasticProblems.............................................. 247<o:p></o:p>11.1Introduction ........................................... 247<o:p></o:p>11.2 LinearElastostatics Problems in 2D ......................... 247<o:p></o:p>11.2.1Basic Theory .................................... 247<o:p></o:p>11.2.2Traction Boundary Conditions ....................... 249<o:p></o:p>11.2.3Fundamental Solutions ............................. 250<o:p></o:p>11.2.4Particular Solutions ............................... 251<o:p></o:p>11.3 HTM,MFS and MPS .................................... 252<o:p></o:p>11.3.1Algorithms of HTM ............................... 252<o:p></o:p>11.3.2Algorithms of MFS and MPS ........................ 252<o:p></o:p>11.4 ErrorsBetween FS and PS ................................ 254<o:p></o:p>11.4.1Preliminary Lemmas ............................... 254<o:p></o:p>11.4.2Polynomials Pn Approximated by xn<o:p></o:p>11.4.3Other Proof for Theorem 11.4.1 ...................... 258<o:p></o:p>11.4.4 ThePolynomials LPn Approximated by Principal FS ...... 261<o:p></o:p>11.5 ErrorBounds for MFS and HTM ........................... 264<o:p></o:p>11.5.1 TheMFS ....................................... 264<o:p></o:p>11.5.2 TheHTM Using FS ............................... 266<o:p></o:p>11.6Numerical Experiments ................................... 268<o:p></o:p>11.7Appendix: Addition Theorems of FS in Linear Elastostatics ....... 271<o:p></o:p>11.7.1Preliminary Lemmas ............................... 271<o:p></o:p>11.7.2Addition Theorems ................................ 277<o:p></o:p>CHAPTER 12<o:p></o:p>CauchyProblems ............................................. 281<o:p></o:p>12.1Introduction ........................................... 281<o:p></o:p>12.2Algorithms of Collocation Trefftz Methods .................... 281<o:p></o:p>12.3Characteristics ......................................... 284<o:p></o:p>12.3.1Existence and Uniqueness ........................... 284<o:p></o:p>12.3.2Ill-Posedness of Inverse Problems ..................... 287<o:p></o:p>12.4 Errorand Stability Analysis ............................... 290<o:p></o:p>12.4.1Error Analysis ................................... 290<o:p></o:p>12.4.2Stability Analysis ................................. 291<o:p></o:p>12.5Applications to Cauchy Data .............................. 295<o:p></o:p>12.5.1Errors on Cauchy Boundary ......................... 295<o:p></o:p>12.5.2Sensitivity of Solutions on Cauchy Data ................ 296<o:p></o:p>12.6Numerical Experiments and Concluding Remarks ............... 297<o:p></o:p>CHAPTER 13<o:p></o:p>3D Problems................................................ 301<o:p></o:p>13.1Introduction ........................................... 301<o:p></o:p>13.2 Methodof Particular Solutions ............................. 302<o:p></o:p>13.3 Methodof Fundamental Solutions ........................... 309<o:p></o:p>VIIIContents<o:p></o:p>13.3.1Algorithms ...................................... 309<o:p></o:p>13.3.2 Linkto MPS..................................... 310<o:p></o:p>13.4 ErrorAnalysis for MFS ................................... 313<o:p></o:p>13.4.1Preliminary Lemmas ............................... 314<o:p></o:p>13.4.2Error Bounds .................................... 321<o:p></o:p>13.5Numerical Experiments ................................... 324<o:p></o:p>13.5.1Collocation Equations on Γ .......................... 324<o:p></o:p>13.5.2 ByMFS ........................................ 325<o:p></o:p>13.5.3 ByMPS ........................................ 330<o:p></o:p>13.6Concluding Remarks ..................................... 331<o:p></o:p>13.7Appendix: 3D Problems of Helmholtz Equations ................ 332<o:p></o:p>13.7.1Interior Dirichlet Problems .......................... 332<o:p></o:p>13.7.2Exterior Dirichlet Problems ......................... 333<o:p></o:p>Part III.Selection of Source Nodes and Related Topics ................ 335<o:p></o:p>CHAPTER 14<o:p></o:p>Comparisonsof MFS and MPS ................................... 339<o:p></o:p>14.1Introduction ........................................... 339<o:p></o:p>14.2 TwoBasis Boundary Methods.............................. 340<o:p></o:p>14.2.1Method of Particular Solutions ....................... 340<o:p></o:p>14.2.2Method of Fundamental Solutions..................... 342<o:p></o:p>14.3 TheMFS-QR .......................................... 346<o:p></o:p>14.3.1Algorithms in Elliptic Coordinates .................... 346<o:p></o:p>14.3.2Characteristics of MFS-QR .......................... 349<o:p></o:p>14.4Numerical Experiments and Comparisons ..................... 354<o:p></o:p>14.4.1Highly Smooth Boundary Data ....................... 355<o:p></o:p>14.4.2Boundary Data with Strong Singularity ................ 356<o:p></o:p>14.4.3Better Pseudo-Boundaries........................... 358<o:p></o:p>14.5Concluding Remarks ..................................... 360<o:p></o:p>CHAPTER 15<o:p></o:p>StabilityAnalysis for Smooth Closed Pseudo-Boundaries ............... 361<o:p></o:p>15.1Introduction ........................................... 361<o:p></o:p>15.2Relations Between FS and PS .............................. 362<o:p></o:p>15.3 Boundsof Cond for Non-Elliptic Pseudo-Boundaries ............. 365<o:p></o:p>CHAPTER 16<o:p></o:p>SingularityProblems from Source Functions; Removal Techniques ........ 375<o:p></o:p>16.1Introduction ........................................... 376<o:p></o:p>16.2Analytical Framework for CTM in [169] ...................... 378<o:p></o:p>16.3 ErrorBounds for Singular Solutions from (16.1.3) ............... 380<o:p></o:p>16.4Singularity for Polygonal Domains and Arbitrary Domains ........ 383<o:p></o:p>16.5Removal Techniques for Laplace’s Equation ................... 384<o:p></o:p>16.5.1 Forthe Case of Q∗ Outside Γ........................ 384<o:p></o:p>16.5.2 Forthe Case of Q∗ Inside Γ under theImage Node Existing . 386<o:p></o:p>16.6Numerical Experiments ................................... 388<o:p></o:p>16.7Applications to Amoeba-Like Domains ....................... 390<o:p></o:p>16.7.1Numerical Results................................. 390<o:p></o:p>16.7.2Removal Techniques Linked to Source Identification<o:p></o:p>Problems............................................ 394<o:p></o:p>16.8Concluding Remarks ..................................... 399<o:p></o:p>CHAPTER 17<o:p></o:p>SourceNodes on Pseudo Radial-Lines .............................. 401<o:p></o:p>17.1Introduction ........................................... 401<o:p></o:p>17.2 PseudoRadial-Lines ..................................... 404<o:p></o:p>17.2.1 OnePseudo Radial-Line ............................ 404<o:p></o:p>17.2.2 TwoPseudo Radial-Lines ........................... 408<o:p></o:p>17.3Stability Analysis ....................................... 409<o:p></o:p>17.3.1Lower Bound Estimates of Cond for Basic Case .......... 409<o:p></o:p>17.3.2Upper Bound Estimates of Cond for Variant Case by Case II. 412<o:p></o:p>17.4Numerical Experiments ................................... 415<o:p></o:p>17.4.1 DiskDomains .................................... 415<o:p></o:p>17.4.2Non-Disk Domains ................................ 420<o:p></o:p>17.5Concluding Remarks ..................................... 424<o:p></o:p>Epilogue.................................................... 427<o:p></o:p>References.................................................. 431<o:p></o:p>Glossary ofSymbols ........................................... 443<o:p></o:p>Index 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