SUMMARYWe study the performance of various upwind techniques implemented in parabolic ÿnite element discretizations for incompressible high Reynolds number ow. The characteristics of an 'ideal' upwind procedure are ÿrst discussed. Then the streamline upwind Petrov=Galerkin method, a simpliÿed version thereof, the Galerkin least squares technique and a high-order derivative artiÿcial di usion method are evaluated on test problems. We conclude that none of the methods displays the desired solution characteristics. There is still need for the development of a reliable and e cient upwind method with characteristics close to those of the 'ideal' procedure.

SUMMARYWe propose inf-sup testing for ÿnite element methods with upwinding used to solve convection-di usion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection e ects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least-squares scheme and a high-order derivative artiÿcial di usion method. The study shows that, as expected, the standard Galerkin method does not pass the inf-sup tests, whereas the other three methods pass the tests. Of these methods, the high-order derivative artiÿcial di usion procedure introduces the least amount of artiÿcial di usion.

We present a 9-node ®nite element for compressible¯ow solutions. A high-order derivative upwind term and a shock capturing term are employed for stability and accuracy of the formulation. We give the solutions of various example problems to illustrate our experiences with the element. Ó

SUMMARYWe propose inf-sup testing for ÿnite element methods with upwinding used to solve convection-di usion problems. The testing evaluates the stability of a method and compactly displays the numerical behaviour as the convection e ects increase. Four discretization schemes are considered: the standard Galerkin procedure, the full upwind method, the Galerkin least-squares scheme and a high-order derivative artiÿcial di usion method. The study shows that, as expected, the standard Galerkin method does not pass the inf-sup tests, whereas the other three methods pass the tests. Of these methods, the high-order derivative artiÿcial di usion procedure introduces the least amount of artiÿcial di usion.

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