This book covers a class of numerical methods that are generally referred to as "Collocation Methods". Different from the Finite Element and the Finite Difference Method, the discretization and approximation of the collocation method is based on a set of unstructured points in space. This "meshless" feature is attractive because it eliminates the bookkeeping requirements of the "element" based methods. This text discusses several types of collocation methods including the radial basis function method, the Trefftz method, the Schwartz alternating method, and the couple collocation and finite element method. Governing equations investigated include Laplace, Poisson, Helmholtz and bioharmonic equations. Regular boundary value problems, boundary value problems with singularity and eigenvalue problems are also examined, Rigorous mathematical proofs are contained in these chapters, and many numerical experiments are also provided to support the algorithms and to verify the theory. A tutorial on the applications of these methods is also provided.
Author(s): Z.-C. Li, T.-T. Lu, H-Y. Hu, A. H.-D. Cheng, null, null
Publisher: WIT
Year: 2008
Language: English
Pages: 433
Cover......Page 1
Trefftz and Collocation Methods......Page 2
Copyright page......Page 5
Contents......Page 8
The bust image of Erich Trefftz (1888–1937)......Page 14
Preface......Page 16
Acknowledgements......Page 20
I.1 Algorithms of CM, TM, and CTM......Page 22
I.2 Coupling techniques......Page 34
I.3 Boundary element methods......Page 42
I.4 Other kinds of boundary methods......Page 46
I.5 Comparisons......Page 49
Part I: Collocation Trefftz method......Page 52
1. Basic algorithms and theory......Page 58
2. Motz's problem and its variants......Page 80
3. Coupling techniques......Page 104
4. Biharmonic equations with singularities......Page 138
Part II: Collocation methods......Page 156
5. Collocation methods......Page 160
6. Combinations of collocation and finite element methods......Page 184
7. Radial basis function collocation methods......Page 208
Part III: Advanced topics......Page 238
8. Combinations with high-order FEMs......Page 242
9. Eigenvalue problems......Page 258
10. The Helmholtz equation......Page 294
11. Explicit harmonic solutions of Laplace's equation......Page 312
Appendix Historic review of boundary methods......Page 344
A.1 Potential theory......Page 345
A.2 Existence and uniqueness......Page 350
A.3 Reduction in dimensions and Green's formula......Page 354
A.4 Integral equations......Page 358
A.5 Extended Green's formula......Page 363
A.6 Pre-electronic computer era......Page 370
A.7 Electronic computer era......Page 377
A.8 Boundary integral equation and boundary element methods......Page 384
References......Page 390
Glossary of symbols......Page 414
C......Page 418
E......Page 419
G......Page 420
L......Page 421
P......Page 423
T......Page 424
Z......Page 425
