In this paper, we propose the element-free Galerkin method based on a polygon support domain since the traditional method cannot enforce the essential boundary condition directly. The proposed method extends the support domain of the computing point to a polygon instead of a rectangle or circular domain, so that the moving least squares shape functions satisfy the property of the Kronecker function, which makes it available for an essential boundary condition to be enforced directly. In addition, the background cells are associated with support domains in the proposed method, which avoids searching nodes tautologically in the traditional method and increases the computational efficiency compared to the traditional method. Numerical examples show that the proposed method not only possesses a higher computational efficiency, but also can successfully conquer numerical instability problems introduced by the dominated convection, when combined with stabilization scheme.