The Method of Fundamental Solutions.

• Superior document: Current Natural Sciences Series
• Author: LI, Zi-Cai.
• TeilnehmendeR: HUANG, Hung-Tsai.; WEI, Yimin.; ZHANG, Liping.
• Place / Publishing House: Les Ulis : : EDP Sciences,, 2023.; ©2023.
• Year of Publication: 2023
• Edition: 1st ed.
• Language: English
• Series: Current Natural Sciences Series
• Physical Description: 1 online resource (472 pages)

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.

Table of Contents

Intro -- The Method of Fundamental Solutions: Theory and Applications -- Contents -- Preface -- Acknowledgements -- Introduction -- Historic Review -- Basic Algorithms -- Numerical Experiments -- Characteristics of the MFS -- Part I -- Dirichlet Problems -- Basic Algorithms of MFS -- Preliminary Lemmas -- Main Theorems -- Stability Analysis for Disk Domains -- Proof Methodology -- Neumann Problems -- Introduction -- Method of Fundamental Solutions -- Description of Algorithms -- Main Results of Analysis and Their Applications -- Stability Analysis of Disk Domains -- Stability Analysis for Bounded Simply-Connected Domains -- Trefftz Methods -- Collocation Trefftz Methods -- Error Estimates -- Concluding Remarks -- Other Boundary Problems -- Mixed Boundary Condition Problems -- Interior Boundary Conditions -- Annular Domains -- Combined Methods -- Combined Methods -- Variant Combinations of FS and PS -- Simplified Hybrid Combination -- Hybrid Plus Penalty Combination -- Indirect Combination -- Combinations of MFS with Other Domain Methods -- Combined with FEM -- Combined with FDM -- Combined with Radial Basis Functions -- Singularity Problems by Combination of MFS and MPS -- Source Nodes on Elliptic Pseudo-Boundaries -- Introduction -- Algorithms of MFS -- Error Analysis -- Preliminary Lemmas -- Error Bounds -- Stability Analysis -- Selection of Pseudo-Boundaries -- Numerical Experiments -- Concluding Remarks -- PartII -- Helmholtz Equations in Simply-Connected Domains -- Introduction -- Algorithms -- Error Analysis for Bessel Functions -- Preliminary Lemmas -- Error Bounds with Small k -- Exploration of Bounded k -- Stability Analysis for Disk Domains -- Application to BKM -- Exterior Problems of Helmholtz Equation -- Introduction -- Standard MFS -- Basic Algorithms -- Brief Error Analysis.

Numerical Characteristics of Spurious Eigenvalues by MFS -- Modified MFS -- Error Analysis for Modified MFS -- Preliminary Lemmas -- Error Bounds -- Stability Analysis for Modified MFS -- Numerical Experiments -- Circular Pseudo-Boundaries by Two MFS -- Non-Circular Pseudo-Boundaries by Modified MFS -- Concluding Remarks -- Helmholtz Equations in Bounded Multiply-Connected Domains -- Introduction -- Bounded Simply-Connected Domains -- Algorithms -- Brief Error Analysis -- Bounded Multiply-Connected Domains -- Algorithms -- Error Analysis -- Stability Analysis for Ring Domains -- Numerical Experiments -- Concluding Remarks -- Biharmonic Equations -- Introduction -- Preliminary Lemmas -- Error Bounds -- Stability Analysis for Circular Domains -- Approaches for Seeking Eigenvalues -- Eigenvalues λk(Φ) and λk(DΦ) -- Bounds of Condition Number -- Numerical Experiments -- Elastic Problems -- Introduction -- Linear Elastostatics Problems in 2D -- Basic Theory -- Traction Boundary Conditions -- Fundamental Solutions -- Particular Solutions -- HTM, MFS and MPS -- Algorithms of HTM -- Algorithms of MFS and MPS -- Errors Between FS and PS -- Preliminary Lemmas -- Polynomials Pn Approximated by (x-e/r2) and(y-n/r2) -- Other Proof for Theorem 11.4.1 -- The Polynomials LPn Approximated by Principal FS -- Error Bounds for MFS and HTM -- The MFS -- The HTM Using FS -- Numerical Experiments -- Appendix: Addition Theorems of FS in Linear Elastostatics -- Preliminary Lemmas -- Addition Theorems -- Cauchy Problems -- Introduction -- Algorithms of Collocation Trefftz Methods -- Characteristics -- Existence and Uniqueness -- Ill-Posedness of Inverse Problems -- Error and Stability Analysis -- Error Analysis -- Stability Analysis -- Trefftz Methods -- Collocation Trefftz Methods -- Applications to Cauchy Data -- Errors on Cauchy Boundary.

Sensitivity of Solutions on Cauchy Data -- Numerical Experiments and Concluding Remarks -- 3D Problems -- Introduction -- Method of Particular Solutions -- Method of Fundamental Solutions -- Algorithms -- Link to MPS -- Error Analysis for MFS -- Preliminary Lemmas -- Error Bounds -- Numerical Experiments -- Collocation Equations on Γ -- By MFS -- By MPS -- Concluding Remarks -- Appendix: 3D Problems of Helmholtz Equations -- Interior Dirichlet Problems -- Exterior Dirichlet Problems -- Part III -- Comparisons of MFS and MPS -- Introduction -- Two Basis Boundary Methods -- Method of Particular Solutions -- Method of Fundamental Solutions -- The MFS-QR -- Algorithms in Elliptic Coordinates -- Characteristics of MFS-QR -- Numerical Experiments and Comparisons -- Highly Smooth Boundary Data -- Boundary Data with Strong Singularity -- Better Pseudo-Boundaries -- Concluding Remarks -- Stability Analysis for Smooth Closed Pseudo-Boundaries -- Introduction -- Relations Between FS and PS -- Bounds of Cond for Non-Elliptic Pseudo-Boundaries -- Singularity Problems from Source Functions -- Removal Techniques -- Introduction -- Analytical Framework for CTM in [169] -- Error Bounds for Singular Solutions from (16.1.3) -- Singularity for Polygonal Domains and Arbitrary Domains -- Removal Techniques for Laplace's Equation -- For the Case of Q* Outside Γ -- For the Case of Q* Inside Γ under the Image Node Existing -- Numerical Experiments -- Applications to Amoeba-Like Domains -- Numerical Results -- Removal Techniques Linked to Source Identification Problems -- Concluding Remarks -- Source Nodes on Pseudo Radial-Lines -- Introduction -- Pseudo Radial-Lines -- One Pseudo Radial-Line -- Two Pseudo Radial-Lines -- Stability Analysis -- Lower Bound Estimates of Cond for Basic Case -- Upper Bound Estimates of Cond for Variant Case by Case II -- Numerical Experiments.

Disk Domains -- Non-Disk Domains -- Concluding Remarks -- Epilogue -- References -- Glossary of Symbols -- Index.