Abstract: The conventional Glalerkin finite element metbod is only applicable to discretizing elliptic equations. In many flow or heat-transfer problems, the governing equations may change type; they become hyperbolic because convection is dominating. The solutions are often corrupted by node-to-node oscillations or ‘wiggles’ the Streamline-Upwind Petrov-Galerkin finite element method can inbibit the crosswind. diffusion and improve the computational accuracy. Two aumerical exarnples are given:the flow of Oldroyd-B fluid around a sphere in a tube and the nonisothermal agitating flow When the Weissenberg number or the Peclet umber is high, the Ga]erlkin method produces,wiggles' while SUPG method yields good results.

Key words: BUPG method, Galerkin method, finite element, hyperbolic