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 coupled collocation and finite element method. Governing equations investigated include Laplace, Poisson, Helmholtz and bi-harmonic 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.