Point interpolation collocation method for the solution of partial differential equations

This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.