Department of Civil Engineering
University of Mississippi

Collocation methods incorporate a wide range of numerical methods that use global, instead of local basis functions for interpolation. Some collocation methods, such as the spectral methods (Chebyshev polynomial, Fourier series, wavelet), require a rectilinear grid, thus are limited to simple geometries. Those methods are not amenable to engineering applications. Other methods, such as the Method of Fundamental Solutions (MFS), the Radial Basis Function (RBF) Collocation, and the Trefftz Method, use scattered points in space for collocation; hence are much more “form‐fitting” to the problem geometry. These methods are also “meshless” because no “elements” are involved. This is a much desired feature for industrial applications, because it cuts down the labor intensive mesh and connectivity data generation. It has driven the industry to develop “meshless FEM”. The real strength of the meshless collocation methods is in its (amazingly) high accuracy. Unlike the more popular numerical methods, such as the Finite Element Method (FEM) and the Finite Difference Method (FDM), which use piece‐wise, low‐degree polynomials for interpolation, collocation methods use global and highly smooth functions. Some interpolants, such as multiquadrics (MQ), are infinitely smooth. As a consequence, some methods, such as the multiquadric or Gaussian collocation method, can achieve exponential error convergence,
 . Using a relatively coarse mesh of 20 20, a uniform accuracy of  has been accomplished solving a Poisson equation (Huang, Lee & Cheng, 2007). This accuracy is impossible to accomplish using FEM or FDM, which typically has an error convergence of  . It will require a mesh refinement and computational effort of astronomical order to achieve this accuracy.