SIAM J. NUMER. ANAL. c 2002 Society for Industrial and Applied Mathematics Vol. 40, No. 1, pp. 254–281
AN ANALYSIS OF SMOOTHING EFFECTS OF UPWINDING STRATEGIES FOR THE CONVECTION-DIFFUSION EQUATION∗
HOWARD C. ELMAN† AND ALISON RAMAGE‡
Abstract. Using a technique for constructing analytic expressions for discrete solutions to the convection-diffusion equation, we examine and characterize the effects of upwinding strategies on solution quality. In particular, for grid-aligned flow and discretization based on bilinear finite elements with streamline upwinding, we show preciselyhow the amount of upwinding included in the discrete operator affects solution oscillations and accuracywhen different types of boundarylayers are present. This analysis provides a basis for choosing a streamline upwinding parameter which also gives accurate solutions for problems with non-grid-aligned and variable speed flows. In addition, we show that the same analytic techniques provide insight into other discretizations, such as a finite difference method that incorporates streamline diffusion and the isotropic artificial diffusion method.
Key words. convection-diffusion equation, oscillations, Galerkin finite element method, stream- line diffusion
AMS subject classifications. 65N22, 65N30, 65Q05, 35J25
PII. S0036142901374877
1. Introduction. There are many discretization strategies available for the lin- ear convection-diffusion equation
(1.1) − ∇2u(x, y)+w ·∇u(x, y)=f(x, y)inΩ, u(x, y)=g(x, y)onδΩ, where the small parameter and divergence-free convective velocity field w =(w1(x, y), w2(x, y)) are given. In this paper, we analyze some well-known methods which in- volve the addition of upwinding to stabilize the discretization for problems involving boundary layers. In particular, we focus on characterizing exactly how this upwinding affects the resulting discrete solutions. A standard discretization technique is the Galerkin finite element method (see, for example, [5], [9], [10], [11], [13]). This is based on seeking a solution u of the weak form of (1.1),