The approximation in the FPM can be summarized as follows. For each point x_i in the analysis domain \Omega (star point), an approximated solution is locally constructed by using a subset of surrounding supporting points x_j, which belong to the problem domain (local cloud of points \Omega_i). The approximation is computed as a linear combination of the cloud unknown nodal values (or parameters) and certain metric coefficients. These are obtained by solving a WLSQ problem at the cloud level, in which the distances between the nodal parameters and the approximated solution are minimized in a LSQ sense. Once the approximation metric coefficients are known, the problem governing PDEs are sampled at each star point by using a collocation method. The continuous variables (and their derivatives) are replaced in the sampled equations by the discrete approximated forms, and the solution of the resulting system allows calculating the unknown nodal values. Hence, the approximated solution satisfying the governing equations of the problem can be obtained. It is important to note that the highly local character of the FPM makes the method suitable for implementing efficient parallel solution schemes.
