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

Abstract

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 Nvert [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 two-dimensional Navier-Stokes equations for an incompressible Newtonian fluid in the case of