Streamline Diffusion Methods for the Incompressible Euler and Navier-Stokes Equations

We present and analyze extensions of the streamline diffusion finite element method to the time-dependent two-dimensional Navier-Stokes equations for an incompressible fluid in the case of high Reynolds numbers. The limit case with zero viscosity, the Euler equations, is also considered. Introduction. The Streamline Diffusion method is a finite element method for convection-dominated convection-diffusion problems recently introduced by Hughes and Brooks [5], [6] in the case of stationary problems. The mathematical analysis of this method for linear problems, together with extensions to time-dependent problems using space-time elements, was started in Johnson and NMvert [8] and was continued in [9], [16] and [10]. The outcome of this work is that the SD (Streamline Diffusion)-method can be demonstrated to have both good stability properties and high accuracy, a combination of desirable features not shared by previously known finite element methods; standard methods either (as the usual Galerkin method) are formally higher-order accurate, but lack in stability and produce severely oscillating solutions if the exact solution is nonsmooth, or (as the classical artificial viscosity or upwind method) contain a large amount of artificial diffusion, limiting the accuracy to at most first-order. The main theoretical results for the SD-method in [8], [9], [16] and [10] are almost-optimal error estimates, together with localization results which show that effects are propagated in the discrete problem in a way similar to what is the case in the continuous problem. In particular, it follows from these localization results that the presence of, e.g., a boundary layer in the exact solution only affects the accuracy of the discrete solution close to the layer. This is in contrast to the usual Galerkin method, where the presence of a boundary layer in general severely degrades the accuracy in the whole domain. The analysis of the SD-method also shows the necessity of sharpening the classical stability concept for finite element (or finite difference) methods for hyperbolic type problems, such as, e.g., convection-diffusion problems with dominating convection. The purpose of this note is to present extensions of the SD-method to some nonlinear hyperbolic problems in fluid mechanics: The time-dependent wo-dimensional Navier-Stokes equations for an incompressible Newtonian fluid in the case of Received November 16, 1984; revised September 23, 1985. 1980 Mathematics Subject Classification. Primary 65N30.