Film cooling technique is currently used in the gas turbine hot sections, such as the combustor wall and the turbine blades, to prevent those sections from failing at elevated temperatures. In the single hole film cooling method, coolant air is injected from a hole into the mainstream and thus the Flow is naturally three dimensional. In this paper, the Navier - Stokes and the energy equations are solved on a flat plate by the finite Element method using cubic elements. Algebraic equations are obtained by the use of the Petrov-Galerkin method. The pressure term is vanished from the momentum equations, by employing the penalty method. The governing equations are transient and the Flow is incompressible and turbulent. The model of turbulence in the near wall region is the wall function method, and in the fully turbulent region is the k-a model. The systems of the algebraic equations are solved by the Frontal method. The film cooling effectiveness data, at the optimum conditions, was directly compared with the experimental results of Goldstein et al. (1968), and a good agreement is demonstrated.