In the most traditional polynomial collocation methods, polynomials are just applied to approximate the solution piecewise around the scattered nodes. These methods with the moving least squares (MLS) approach are unstable and usually can work only for problems with simple boundary conditions and regular geometry. Polynomials have been rarely used globally such as the basis functions being treated in the radial basis functions collocation methods. Weak form methods with MLS approach, which need background cells for local integration, perform with higher stability and are applied more widely in many numerical im- plementations although they are sometimes unstable. Study on improving the collocation techniques and the construction of local clouds, which means the selection of neighboring nodes involved in the local approximation, is important issue for increasing stability. Besides the aforementioned meth- ods, one can also find a few literatures about meshless methods employing the reproducing kernel approximation. These methods with complicated formulation accurately approximate solutions of some partial differential equations as a linear combination of the shape functions in a global way. Although in the polynomial collocation methods with MLS approach the global solution is also the linear combination of the shape functions, the basic concept differs from the reproducing kernel ap- proximation so that the way for collocation is not the same. In methods with reproducing kernel approximation, such as finite cloud methods, the partial derivatives of the solution are obtained from the differentiation of the global shape functions, whereas in polynomial collocations methods with MLS approach, they are obtained by differentiating the local basis functions. In this thesis, a robust local polynomial collocation method which is based on the strong weak form is dealt with for solving some time-independent problems.This method is similar to the finite point method in which polyno- mials are localized by putting their origins at the collocation points. On the basis of the collocation approach, this method is as simple and straightforward as other analogous methods. The satisfaction of the governing equation is additionally required on the boundary collocation points in this newly proposed polynomial collocation method.Therefore, the method is developed in a way that not only the governing equation is satisfied on the boundaries but also the problem boundary conditions are satisfied. The sensitivity of the shape parameter, the local supporting range of the shape functions in the MLS approach and the convergence of the nodal resolution are studied by solving some test problems. Proposed method is also compared with some conventional methods such as fixed kernel and Hermit approximations. The robust local polynomial collocation method is further verified by applying it to a steady state convection diffusion problem. Finally, the present method is applied to calculate the velocity fields of two potential flow problems. Results obtained show that this method is more accurate and robust than the conventional collocations methods especially in estimating the partial derivatives of the solution near the boundary. In fact, accurate partial derivatives of the solu- tion can be obtained in the process of seeking the solution.This method can be further developed for solving not only some complicated problems but also some time-dependent problems.