The numerical simulation for the exterior problem of Poisson equation is considered. We introduced a polygonal artificial boundary Γe and designed a discrete artificial boundary condition on it by using the direct method of lines. Then the original problem is reduced to a boundary value problem defined in a bounded computational domain with a polygonal boundary. The finite element approximation of this reduced boundary value problem is considered and it is proved that the finite element approximate problem is well posed. Furthermore numerical examples show that the discrete artificial boundary condition is very effective and more accurate than the Neumann boundary condition which is often used in engineering literatures.