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For a model convection-dominated singularly perturbed convection-diffusion prob- lem, it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion. The ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5/4 under local smoothness assumptions. 1. Introduction. The streamline diffusion method is a finite element method for convection-dominated convection-diffusion problems which combines formal high accuracy with decent stability properties. The method was introduced in the case of stationary problems by Hughes and Brooks (7), cf. Raithby and Torrance (14) and Wahlbin (15) for earlier thoughts in this direction. The mathematical analysis of the method was started in Johnson and Navert (8) and continued with extensions to, e.g., time-dependent problems in Navert (12), Johnson, Navert and Pitkaranta (9) and Johnson and Saranen (10). In these papers local error estimates in L2 of order O(h k 1/2), in regions of smoothness, with piecewise polynomial finite elements of degree k, were derived, together with estimates stating, as a typical example, that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0(h1/2), essentially. The purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0(h3/4). The improvement from 0(h1/2) to 0(h3/4) is obtained by adding a small amount, 0(h3/2), of artificial crosswind diffusion to the method. In the piecewise linear case (k = 1) this does not destroy the known O( h3/2) accuracy in L2 in smooth regions. Using our first result, we then obtain our second main result, localized pointwise error estimates of order 0(h5/4) in regions of smoothness. (The previously known best pointwise error estimate in the piecewise linear situation is 0(h1/2).) Another consequence is a global L,-estimate of order 0(h'/2) in the presence of typical crosswind and downwind singularities. We shall consider the model problem of finding u = u(x, y) such that

Crosswind smear and pointwise errors in streamline diffusion finite element methods