Localized collocation schemes and their applications

This paper presents a summary of various localized collocation schemes and their engineering applications. The basic concepts of localized collocation methods (LCMs) are first introduced, such as approximation theory, semianalytical collocation methods and localization strategies. Based on these basic concepts, five different formulations of localized collocation methods are introduced, including the localized radial basis function collocation method (LRBFCM) and the generalized finite difference method (GFDM), the localized method of fundamental solutions (LMFS), the localized radial Trefftz collocation method (LRTCM), and the localized collocation Trefftz method (LCTM). Then, several additional schemes, such as the generalized reciprocity method, Laplace and Fourier transformations, and Krylov deferred correction, are introduced to enable the application of the LCM to large-scale engineering and scientific computing for solving inhomogeneous, nonisotropic and time-dependent partial differential equations. Several typical benchmark examples are presented to show the recent developments and applications on the LCM solution of some selected boundary value problems, such as numerical wave flume, potential-based inverse electrocardiography, wave propagation analysis and 2D phononic crystals, elasticity and in-plane crack problems, heat conduction problems in heterogeneous material and nonlinear time-dependent Burgers’ equations. Finally, some conclusions and outlooks of the LCMs are summarized.