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.

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