Trefftz and Collocation Methods

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 methods, 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, and the coupled collocation and finite element method. Governing equations investigated include Laplace, Poisson, Helmholtz and biharmonic 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.

Table of Contents:
Tutorial introduction Algorithms of CM, TM, and CTM; Coupling techniques; Boundary element methods; Other kinds of boundary methods; Comparisons Part I: Collocation Trefftz method 1 - Basic algorithms and theory Notations and preliminaries; Approximation problems; Error estimates; Debye-Huckel equation; Stability analysis 2 - Motz's problem and its variants Introduction; Basic algorithms of CTM; Error bounds and integration approximation; Variants of Motz's problem; Concluding remarks 3 - Coupling techniques Introduction; Description of generalized TMs; Penalty TMs; Simplified hybrid TMs; Penalty plus hybrid TMs; Lagrange multiplier TM; Effective condition number; Numerical experiments 4 - Biharmonic equations with singularities Introduction; The Green formulas of A2u; The collocation Trefftz methods; Error bounds Part II: Collocation methods 5 - Collocation methods Introduction; Description of collocation methods; Error analysis; Robin boundary conditions; Inverse inequalities; Final remarks 6 - Combinations of collocation and finite element methods Introduction; Combinations of FEMs; Linear algebraic equations of combination of FEM and CM; Uniformly Vh0-elliptic inequality; Uniformly Vh0-elliptic inequality involving integration approximation; Final remarks 7 - Radial basis function collocation methods Introduction; Radial basis functions; Description of radial basis function collocation methods; Inverse estimates for radial basis functions; Numerical experiments; Comparisons and conclusions Part III: Advanced topics 8 - Combinations with high-order FEMs Introduction; Combinations of TM and Lagrange FEMs; Global superconvergence; Adini's elements 9 - Eigenvalue problems Introduction; New numerical algorithms for eigenvalue problems; Error bounds of eigenvalues; Error bounds of eigenfunctions; Computational models and numerical experiments; Eigenvalues for the singularity problem; Summaries and discussions 10 - The Helmholtz equation Introduction; The Trefftz method; Error analysis; Summaries and discussions 11 - Explicit harmonic solutions of Laplace's equation Introduction; Harmonic functions; Harmonic solutions involving Neumann conditions; Extensions and analysis on singularity; New models of singularities for Laplace's equation; Concluding remarks Appendix - Historic review of boundary methods Potential theory; Existence and uniqueness; Reduction in dimensions and Green's formula; Integral equations; Extended Green's formula; Pre-electronic computer era; Electronic computer era; Boundary integral equation and boundary element methods

About the Author :
Zi-Cai Li (Ph.D., University of Toronto, Canada) is a professor in the Department of Applied Mathematics at National Sun Yat-Sen University, Taipei, Taiwan. His research specialties are Numerical Analysis, Scientific Computation, Image Processing, and Pattern Recognition. Tzon-Tzer Lu (Ph.D., University of California at Berkley,USA) is a researcher in the Mathematics Division of the National Center for Theoretical Science, Taiwan. [note: he's not listed on the website; I'm just going by the author info supplied] Hsin-Yun Hu (Ph.D., National Sun Yat-Sen University, Taiwan) is a Post-Doctoral Fellow in the Department of Mathematics at National Tsin Hua University, Taiwan. Dr. Alexander H.-D. Cheng has been Professor and Chair of the Department of Civil Engineering at the University of Mississippi since 2001having previously taught at University of Delaware, Columbia University, and Cornell University. His research interests include water resources, boundary element methods, mechanics of porous materials, and ocean acoustics. He is currently Editor of the international journal Engineering Analysis with Boundary Elements (Elsevier), Associate Editor of the Journal of Engineering Mechanics (ASCE), and Editor-in-Chief of the book series Progress in Water Resources (WIT Press). Dr. Cheng is the author of the book Multilayered Aquifer Systems (Dekker, 2000), and the editor of a dozen books, including Computational Stochastic Mechanics (CMP/Elsevier 1993), Saltwater Intrusion in Coastal Aquifers (Kluwer, 1999), Environmental Fluid Mechanics (ASCE, 2002), and Coastal Aquifer Management--Monitoring, Modeling and Recent Practices (CRC Press, forthcoming). He has authored more than one hundred refereed journal papers. He has chaired a number of international conferences, including two conference series that he co-founded: the First International Conference on Saltwater Intrusion and Coastal Aquifers (2001) and the First Biot Conference on Poromechanics (1998). He was awarded the Walter L. Huber Civil Engineering Research Prize by ASCE in 1994, the Basic Research Award by the U.S. National Committee for Rock Mechanics, National Research Council in 1994 and 1999, and the Eminent Scientist Award, Wessex Institute of Technology in 2001. He is an Honorary Professor of the Department of Engineering, Kings College, University of Aberdeen, and a Fellow of the Wessex Institute of Technology, UK. He serves as Chair of Executive Committee of Engineering Mechanics Division, ASCE, and the Vice President for Academic Affairs of the American Institute of Hydrology. He is cited in Who's Who in the World, Who's Who in America, Who's Who in Science and Engineering, and American Man and Woman of Science.