Recently, the developments of the so-called meshfree or meshless methods have been more attractive. This kind of methods only use a set of nodes scattered within the problem domain as well as a set of nodes scattered on the boundary. Therefore, they have many advantages over the conventional numerical methods. At present, there are many meshfree methods such as element-free Galerkin (EFG) method, meshless local Petrov–Galerkin (MLPG) method, reproducing kernel particle method (RKPM) and so on. In the mid-90s Hughes revisited the origins of the stabilization schemes from a variational multiscale point of view and presented the variational multiscale (VM) method. Recently, Zhang et al. followed the idea of the variational multiscale finite element method (VMFEM) and proposed the variational multiscale EFG (VMEFG) method and the two-level EFG method respectively for some benchmark problems in the fields of fluid and magnetic. However, they still assumed that the fine scale vanished identically over the element boundaries although non-zero within the background integral elements, which should not be suitable. Besides the research of Zhang and Yeon et al. in order to take full advantage of the meshfree methods and avoid the strong assumption mentioned above, zhang’s paper proposes a new coupling between VM and meshfree methods for 2D Burgers’ equation. A new numerical method, which is based on the coupling between variational multiscale method and meshfree methods, takes full advantage of the meshfree methods and therefore, no mesh generation and mesh recreation are involved. One of the major features of EFG and VMEFG methods is that the shape functions are formed by the application of the moving least-square (MLS) approximation. Therefore the approximated field function is continuous and smooth in the entire computed domain. In the first chapter of the thesis, some preliminaries have been gathered. In the second chapter after dealing with the MLS method, two problems which have exact solutions are solved to analyze the convergence behavior of the proposed method. The Burgers’ equation is a useful model for many interesting physical problems, such as shock wave, acoustic transmission, traffic and aerofoil flow theory, turbulence and supersonic flow as well as a prerequisite to the Navier–Stokes equations. This equation can be considered as an evolutionary process in which a convective phenomenon is in contrast with a diffusive phenomenon. When diffusive phenomenon is dominative, the Burgers’ equation is parabolic, or else it is hyperbolic. That is to say, this equation usually exhibits a ‘‘mixed” property. The 1D Burgers’ equation having large Reynolds number is solved in the second chapter. At the end of this chapter a 2D Burgers’ equation having large Reynolds number is solved. The numerical results show that the proposed method can indeed obtain accurate numerical results for the equation having large Reynolds number, which does not refer to the choice of a proper stabilization parameter. The nonlinear Schr?dinger equation (NLS) and the sine-Gordon equation describe many physical phenomena and have important applications in fluid dynamics, nonlinear optics, plasma physics, fiber optics, dislocation theory and Josephson physics. These equations are solved in the third and fourth chapters, respectively. In comparison with the element free Galerkin method, Variational multiscale element free Galerkin method can often generate more stable numerical solutions.