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 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.

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.



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.