A new numerical approach is proposed to solve the equations of flow and mass transport. This approach relies on a finite-element type formulation using basis functions of arbitrary order (spectral elements). An appropriate choice of the nodal points and the integration points leads to a very effective computation of the element matrices. The transport equation is solved by combining a spectral approximation for the dispersion term with a backtracking algorithm for the advection term, which is aided by the most precise interpolation possible with the spectral method. The formulation is shown to yield an accurate description of very steep fronts with reasonable computer resources. ¿ American Geophysical Union 1993