Abstract: Element Free Galerkin(EFG) method is intensive computation due to the requirement of background cells quadrature, and may appear nonphysical oscillations in solving convection dominated problems. To solve these problems, a local Taylor expansion integration ElementFree Galerkin (LTEIEFG) method is proposed in this paper, which based on the idea of integration for local Taylor expansion. This method can not only save computation cost but also eliminate nonphysical oscillation in solving convection dominated problem. Numerical examples of one-dimensional convection diffusion equation and two-dimensional Burgers equation show that the LTEIEFG method is accurate and much more efficient than EFG method.