Summary: We investigate the optimal accuracy of the streamline diffusion finite element method applied to convection-dominated problems. For linear/bilinear elements the theoretical order of convergence given in the literature is either \(O(h^{3/2})\) for quasi-uniform meshes or \(O(h^2)\) for some uniform meshes. The determination of the optimal order in general was an open problem. By studying a special type of meshes, it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes.