The Method of Fundamental Solutions.

The Method of Fundamental Solutions.

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 (B...

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Bibliographic Details
Superior document:Current Natural Sciences Series
:
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)
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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.

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