The authors describe a new finite element method for the solution of convection-dominated diffusion equations. In detail they study the case of bilinear elements on rectangles. Because of the well-known instability of symmetric finite elements on an equidistant mesh they suggest to extend the space of test functions by local biparabolic functions in order to suppress unwanted oscillatory solutions. The stability effect of this extension is explained by functional analytical methods. As a drawback there results an overdetermined system of linear equations, which is solved in least-squares sense using the normal equations. To describe the accuracy of their scheme, the authors prove that the method is nodally exact for all solutions which are polynomials of degree at most two. With numerical illustrations they compare the results with the results obtained by the streamline upwind/Petrov-Galerkin method proposed by \textit{A. N. Brooks} and \textit{Th. J. R. Hughes} [Comput. Meth. Appl. Mech. Engrg., 32, 199-259 (1982; Zbl 0497.76041)].