The Method of Fundamental Solutions: Theory and Applications

Name: The Method of Fundamental Solutions: Theory and Applications
Brand: EDP sciences
Price: 362.08 EUR
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Zi-cai Li Hung-Tsai HUANG Yimin Wei Liping Zhang(Autor*in)

EDP sciences (Verlag)

Erschienen am 4. Dezember 2023

470 Seiten

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978-2-7598-3172-2 (ISBN)

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Inhalt

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 ?k(F) and ?k(DF)
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 G
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 G
For the Case of Q* Inside G 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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